The user wants a single-file HTML calculator and SEO article for “calculating limits using the limit laws”.

**Analysis of the Request:**

1. **Topic:** Calculating limits using limit laws (Calculus).
2. **Role:** Senior SEO expert and frontend developer.
3. **Format:** Single HTML file (from `` to ``).
4. **Tech Stack:** HTML, CSS (embedded), JavaScript (embedded).
* **JS Constraints:** `var` only, no `const`, no `let`, no arrow functions, no template literals, no external libraries.
5. **Layout:** Single-column, max-width 960-1100px, centered.
6. **Visual Style:** Professional, #004a99 primary, #28a745 success, #f8f9fa background.
7. **Calculator Requirements:**
* Topic-specific inputs (e.g., Function $f(x)$, Value $a$, approaching value).
* Real-time updates.
* Reset button.
* Copy results button.
* **Tables & Charts:** At least one table, at least one dynamic chart (Canvas or SVG) with 2+ series, updating dynamically.
* Mobile responsive tables/charts.
8. **SEO Article Requirements:**
* Deep, long-form.
* Specific sections: Definition, Formula/Explanation + Variables Table, Practical Examples, How to Use, Key Factors, FAQ (8+), Related Tools.
* Keyword density >= 4% for “calculating limits using the limit laws”.
* Internal linking (at least 6 links).
* Semantic HTML.
* One H1.

**Calculator Logic Design:**

Since this is a math topic, a “Limit Calculator” is the most appropriate tool.
* **Inputs:**
* Function $f(x)$ (Text input, e.g., “x^2 + 2x”)
* Approach value $L$ (Number input, e.g., 3)
* Direction (Left, Right, Two-sided) – *Actually, limit laws usually imply two-sided unless specified, but for a calculator, we might just evaluate the limit at a point.*
* Let’s simplify: Input a function string and a value to approach.
* **Calculation Logic (JavaScript):**
* Parsing a mathematical function string in vanilla JS without libraries is complex. I will implement a basic parser or use `Function` constructor (carefully) to evaluate the function at points close to the limit.
* *Approach:* To demonstrate “Limit Laws”, the calculator should probably show the step-by-step application or at least evaluate the limit by sampling values close to $a$.
* *Better approach for a “Limit Law” calculator:* Allow the user to input components of a sum/product/quotient to see how the laws apply, OR simply calculate the limit of a polynomial/rational function.
* *Decision:* Let’s build a “Limit Evaluator”. User inputs $f(x)$ and $x \to a$. The calculator evaluates $f(x)$ at $a-0.1, a-0.01, a+0.01, a+0.1$ to approximate the limit and display the result.
* *Refinement:* To make it educational about “Limit Laws”, maybe inputs for $f(x)$ and $g(x)$ and an operation? No, that’s too specific. Let’s stick to a general Limit Calculator that evaluates $\lim_{x \to a} f(x)$.
* *Parsing:* I’ll use a safe-ish evaluation method. Since I can’t use libraries, I’ll replace `^` with `**` and use the `Math` object context.

**SEO Content Strategy:**

* **Primary Keyword:** “calculating limits using the limit laws”, “limit laws calculator”.
* **Structure:**
* H1: Limit Laws Calculator & Guide
* H2: What are Limit Laws?
* H2: The Limit Laws Formula
* H2: How to Use This Calculator
* H2: Examples of Calculating Limits
* H2: Key Factors in Limit Evaluation
* H2: Frequently Asked Questions
* **Internal Links:** Placeholders for “Calculus Tools”, “Derivative Calculator”, “Integral Calculator”, etc.

**Drafting the Code Structure:**

* **HTML:**
* Header
* Main Container
* Calculator Section (`.date-calc-container`)
* SEO Article Section
* **CSS:**
* Responsive, single column.
* Table scrolling.
* Canvas responsive.
* **JS:**
* `calculateLimit()` function.
* `updateChart()` function.
* `copyResults()` function.
* `resetCalculator()` function.
* Event listeners on inputs.

**Specific Calculator Logic (The tricky part without libraries):**
I will implement a basic expression evaluator.
1. Get string $f(x)$.
2. Get value $a$.
3. Evaluate $f(a – \epsilon)$ and $f(a + \epsilon)$ for small $\epsilon$ (e.g., 0.0001).
4. If they are close, that’s the limit.
5. If $f(a)$ is defined and continuous, return $f(a)$.
6. *Constraint Check:* The prompt asks for “calculating limits using the limit laws”. A calculator that just spits out a number doesn’t teach the laws.
* *Pivot:* Let’s make the calculator handle specific cases: Sum, Product, Quotient.
* *Inputs:*
* Function 1: $f(x) = $ (e.g., $2x$)
* Limit of f(x) as $x \to a$: (User enters this, or calculator computes it? Let’s compute it).
* Function 2: $g(x) = $ (e.g., $x^2$)
* Limit of g(x)…
* Operation: Sum, Product, Quotient.
* *Actually, the prompt says “calculating limits using the limit laws”.* The most useful tool is one that calculates the limit of a function. I will stick to a general Limit Calculator but display the “Law” being used if possible, or simply focus on the calculation aspect as requested by the “calculator functionality” rules (primary result, intermediate values).
* Let’s stick to: Input $f(x)$, Input $x \to a$. Output: Limit value, Left-hand limit, Right-hand limit. This demonstrates the Squeeze Theorem or Direct Substitution property (Law 1).

**JavaScript Implementation Details:**
* Use `try…catch` block for evaluation.
* Replace `sin` with `Math.sin`, etc.
* Handle `^` to `**`.
* Chart: Plot $y = f(x)$ near $x = a$. Show the point approaching the limit.

**Refining the “Limit Laws” aspect:**
Since I cannot build a full CAS (Computer Algebra System) in vanilla JS in a single file easily, I will focus on the **Direct Substitution Law** and **Evaluation**.
The calculator will:
1. Take $f(x)$ and $a$.
2. Calculate $f(a)$ (if continuous).
3. Calculate $f(a – 0.001)$ and $f(a + 0.001)$.
4. Display these as intermediate values to show the “approach”.
5. Result is the limit.

**Let’s write the code.**

* **Inputs:**
* `funcInput`: String (e.g., `(x^2 – 1)/(x – 1)`)
* `limitPoint`: Number (e.g., `1`)
* `epsilon`: Number (for chart scale, default 2)
* **Outputs:**
* `result`: The calculated limit.
* `leftVal`: $f(a – 0.0001)$
* `rightVal`: $f(a + 0.0001)$
* `evalVal`: $f(a)$ (if defined)

**SEO Keywords:**
* calculating limits using the limit laws
* limit laws calculator
* calculus limits
* how to calculate limits

**Internal Links (Placeholders):**
* /tools/derivative-calculator/
* /tools/integral-calculator/
* /learn/calculus-basics/
* /learn/continuity-theorem/
* /tools/fraction-calculator/
* /blog/math-tips/

**Step-by-Step Implementation Plan:**

1. **HTML Skeleton:** Standard HTML5.
2. **CSS:** Professional styling, responsive.
3. **Calculator UI:**
* Input for Function $f(x)$.
* Input for approaching value $a$.
* Calculate button (or auto-calc).
* Reset button.
* Result display area.
* Canvas for graphing.
* Table for values.
4. **JS Logic:**
* `evaluateFunction(str, x)`: Parses string, handles `^`, `sin`, `cos`, etc.
* `calculate()`: Main logic.
* `drawChart()`: Plots the function.
5. **Article Content:** Write the long-form text based on the requirements.

**Self-Correction/Constraint Check:**
* *Constraint:* “Use `var` ONLY”. Checked.
* *Constraint:* “Single-column”. Checked.
* *Constraint:* “No markdown fences”. Checked.
* *Constraint:* “Mobile responsive tables/charts”. Checked.

**Code Construction (Mental Draft):**
* *Parser:* `return new Function(‘x’, ‘return ‘ + sanitizedStr)(val)` is the easiest way, but requires standard JS syntax (e.g., `Math.sin(x)`). I will add a helper to replace `sin(x)` with `Math.sin(x)` etc., to make it user-friendly.

**Final Polish:**
Ensure the tone is professional and educational. The calculator should look like a tool for a math student or engineer.

Let’s generate the HTML.

**Variables:**
* Primary Keyword: calculating limits using the limit laws
* Related Keywords: limit laws, limit calculator, calculus limits, evaluate limits, direct substitution
* Internal Links: /tools/limit-calculator, /tools/derivative-calculator, /tools/integral-calculator, /learn/calculus-intro, /learn/continuity, /tools/scientific-calculator

**Article Structure:**
1. H1: Mastering the Art of Calculating Limits Using the Limit Laws
2. Intro: Brief overview.
3. Section: What are Limit Laws?
4. Section: The Mathematics of Limits (Formula/Table)
5. Section: How to Use This Calculator
6. Section: Practical Examples
7. Section: Factors Affecting Limits
8. Section: FAQ
9. Section: Related Tools

Let’s go.

**Note on JS Math Parsing:**
To make `x^2` work, I need to replace `^` with `**`.
To make `sin(x)` work, I need to map `sin` to `Math.sin`.
I will implement a robust enough parser for standard polynomials and rational functions.

**Chart:**
Canvas element.
X-axis: from $a – 2$ to $a + 2$.
Y-axis: dynamic based on values.
Plot the function curve.
Draw a vertical line at $x=a$.
Draw points at $a \pm \delta$.

**Table:**
Columns: Approach Method, x value, f(x) value.
Rows: Left-hand, Right-hand, Direct Substitution.

**Copy Results:**
Format: “Limit of [func] as x approaches [a] is [result]. \n\nDetails: \nLeft: … \nRight: …”

Ready to write.

**One specific detail:** The prompt asks for “calculating limits using the limit laws”. While I can’t code a symbolic algebra system that *shows* the steps of applying Sum Law, Constant Law, etc., I can frame the calculator as an implementation of the **Direct Substitution Property** (which is a consequence of continuity, often taught alongside limit laws) and show the numerical verification of the limit, which is the practical application of the definition. I will label the calculator as a “Limit Laws Calculator & Evaluator”.

Let’s execute.

**CSS Colors:**
Primary: #004a99
Success: #28a745
Bg: #f8f9fa

**HTML Structure:**

… meta tags …
Limit Laws Calculator & Guide | Calculate Limits Online

…

**JS Logic for `var`:**
`var funcStr = …`
`var limitA = …`
`var result = …`

**Handling `sin`, `cos`, `tan`:**
Regex replace `/\b(sin|cos|tan|log|sqrt|exp)\b/g` to `Math.$1`.

**Handling `^`:**
`str.replace(/\^/g, ‘**’)`

**Handling implicit multiplication (e.g. `2x`):**
This is hard without a library. I will stick to explicit JS syntax or simple `2*x` but add a helper note. Or, I can try a regex for `(\d)([a-z])` -> `$1*$2`. I’ll add basic support for that.

**Final check on “Limit Laws” specific inputs:**
Maybe inputs for $f(x)$ and $g(x)$?
No, a single function limit is more useful for a general tool. I will stick to the single function evaluator but explain the laws in the text.

Let’s produce the output.

**Keyword Density Check:**
“calculating limits using the limit laws” needs to appear frequently.
I will use synonyms like “limit laws”, “evaluating limits”, “limit properties” to keep it natural but hit the density.

**Internal Links:**
I will use `…` tags with the provided placeholders.

**Responsive Chart:**
``
CSS: `canvas { max-width: 100%; height: auto; }`

**Responsive Table:**
`

…

Let’s write the code.

**Input IDs:**
`funcInput`
`limitPoint`
`epsilonInput` (optional, maybe just fixed or derived)

**Output IDs:**
`mainResult`
`leftResult`
`rightResult`
`directResult`
`resultTable`
`limitChart`

**Copy Button:**
`onclick=”copyResults()”`

**Reset Button:**
`onclick=”resetCalc()”`

**Real-time:**
`oninput=”calculate()”` on inputs.

**Validation:**
Check if function is valid. If `NaN`, show error.

**Error Handling:**
If `try/catch` fails, show “Invalid Function” error message below input.

**Helper Text:**
“Enter x as ‘x’. Use * for multiplication. Ex: 2*x + 3.”

Let’s build it.

**Article Content Generation:**
* **Intro:** Calculating limits using the limit laws is a fundamental skill in calculus.
* **Section A:** Definition. Limit laws are rules for evaluating limits.
* **Section B:** Formula. $\lim_{x \to a} [f(x) \pm g(x)] = \lim f \pm \lim g$.
* **Section C:** Examples. $\lim_{x \to 2} (3x+1)$.
* **Section D:** How to use.
* **Section E:** Factors. Continuity, domain, form (0/0).
* **Section F:** FAQ.
* **Section G:** Links.

Okay, ready.

**One detail on the prompt:** “Input section: Main wrapper class: `.date-calc-container`”.
I will use this class name even though it’s a math calculator, to adhere to the strict formatting rules provided.

**One detail on the prompt:** “Ensure your document.getElementById(“ID”) calls MATCH EXACTLY the formatting of “.
I will be very careful with casing.

**One detail on the prompt:** “Use inline onclick events”.
I will use `onclick=”…”` for buttons.

**One detail on the prompt:** “NO markdown fences”.
Output will be raw HTML.

**One detail on the prompt:** “Use `var` ONLY”.
I will scan the JS to ensure no `const` or `let`.

Let’s go.

**Chart Logic:**
I need to generate data points.
Loop from `a – range` to `a + range` with step `0.1`.
Draw lines.
Draw the limit point.

**Table Logic:**
Row 1: Left-hand limit ($a – 0.001$)
Row 2: Right-hand limit ($a + 0.001$)
Row 3: Direct substitution ($a$)

**Final check on “Limit Laws” in the calculator:**
The calculator effectively uses the **Law of Direct Substitution** (if continuous) or the **Definition of Limit** (numerically) to find the result. I will label the result “Calculated Limit (using Numerical Approximation)”.

Let’s write the HTML.

**Variables for Article:**
Primary: calculating limits using the limit laws
Related: limit laws, limit calculator, calculus limits, evaluate limits, direct substitution property, infinite limits
Internal: /tools/limit-calculator, /tools/derivative-calculator, /tools/integral-calculator, /learn/calculus-101, /learn/continuity, /tools/scientific-calculator

**Article Text Drafting (Mental):**
“Calculating limits using the limit laws allows mathematicians to break down complex functions…”
“The Limit Laws Calculator on this page helps you verify your manual calculations…”

Okay, proceeding to generate the code.

**Input Validation:**
If `funcInput` is empty, clear results.
If `limitPoint` is empty, clear results.

**Helper Text:**
“Supported: +, -, *, /, ^, sin, cos, tan, log, sqrt, exp. Use x as variable.”

**Colors:**
Primary: #004a99
Success: #28a745

**Font:**
System fonts (Arial, sans-serif).

**Layout:**
Centered, max-width 1000px.

**Mobile:**
Padding 15px.

**Chart:**
Canvas.

**Table:**
Standard HTML table.

**Copy:**
Clipboard API `navigator.clipboard.writeText`.

**Reset:**
Set values to default.

**Final Code Structure:**
``
``
``… styles …``
``
`

`
`

`…`

`
`

`… inputs …`

`
`

`… results …`

`
`

`… canvas …`

`
`

`… table …`

`
`

`… content …`

`
``
``
``

**Specific JS Logic for `var`:**
`var x = …`
`var result = …`
`function calculate() { … }`
`function draw() { … }`

**Handling `^`:**
`str = str.replace(/\^/g, ‘**’);`

**Handling implicit mult:**
`str = str.replace(/(\d)([a-z\(])/g, ‘$1*$2’);` (Simple regex, might have edge cases but works for basic inputs).

**Handling Math functions:**
`str = str.replace(/\b(sin|cos|tan|log|sqrt|exp|abs|pow)\b/g, ‘Math.$1’);`

**Safety:**
Wrap evaluation in `try…catch`.

**Chart Drawing:**
`ctx.beginPath()`, `ctx.moveTo()`, `ctx.lineTo()`, `ctx.stroke()`.

**Responsive Chart:**
On window resize? Or just CSS `max-width: 100%`. CSS is easier and sufficient.

**Final check on “Limit Laws” specific requirement:**
The prompt asks for “calculating limits using the limit laws”.
I will add a section in the results called “Applied Limit Laws” where I list:
1. Constant Law
2. Identity Law
3. Sum Law
… and say “Applied to function…”.
Actually, without symbolic parsing, I can’t know *which* laws were used.
I will stick to the numerical verification approach which *demonstrates* the result of the laws.

Let’s write.

**One last check on “Input section” requirements:**
“Main wrapper class: `.date-calc-container`” -> Check.
“Each input wrapped in `.input-group`” -> Check.
“Inline validation ONLY (no alerts)” -> Check.
“Error messages shown directly below the input” -> Check.
“Reset button” -> Check.
“Copy Results button” -> Check.

**One last check on “Results behavior”:**
“Results update in real time when inputs change” -> `oninput` event.
“Copy Results button” -> Check.

**One last check on “Tables & Charts”:**
“At least one structured table” -> Check.
“At least one dynamic chart” -> Check.
“No external chart libraries” -> Check.
“Chart must include at least two data series” -> I will plot the function curve and the limit point (or two different epsilon approaches).
“Chart must update dynamically” -> Check.

**Series 1:** The function curve $f(x)$.
**Series 2:** The limit point (visualized as a dot or line).

Let’s do this.

**Input IDs:**
`funcInput`
`limitPoint`

**Output IDs:**
`mainResult`
`leftResult`
`rightResult`
`evalResult`
`errorMsg`
`resultTable`
`limitChart`

**Helper Text:**
“Enter function in terms of x. Example: (x^2 – 4) / (x – 2)”

**CSS:**
`.input-group { margin-bottom: 20px; }`
`.error-msg { color: red; font-size: 0.9em; display: none; }`
`.result-box { background: #e9ecef; padding: 15px; border-radius: 5px; }`

**SEO Article:**
Must be below calculator.
Must use H2, H3.
Must have internal links.

**Internal Links:**
1. /tools/derivative-calculator
2. /tools/integral-calculator
3. /learn/calculus-basics
4. /learn/continuity-theorem
5. /tools/scientific-calculator
6. /blog/math-tutorials

**Keyword Density:**
“calculating limits using the limit laws” -> ~15 times in the text.

**Final Plan:**
1. Write HTML structure.
2. Write CSS.
3. Write JS (Parser, Calc, Chart, Copy, Reset).
4. Write Article Content.

**Constraint Checklist & Confidence Score:**
1. Single HTML file? Yes.
2. CSS embedded? Yes.
3. JS embedded? Yes.
4. `var` only? Yes.
5. Single-column? Yes.
6. Professional Date style? Yes.
7. Responsive? Yes.
8. Calculator specific topic? Yes.
9. Real-time? Yes.
10. Reset/Copy? Yes.
11. Table/Chart? Yes.
12. SEO Article? Yes.
13. Internal Links? Yes.
14. No Markdown? Yes.

Confidence Score: 5/5

**Mental Sandbox Simulation:**
* User types `(x^2 – 1)/(x – 1)` into `funcInput`.
* User types `1` into `limitPoint`.
* JS replaces `^` -> `**`.
* JS evaluates at `0.999` -> `1.999`.
* JS evaluates at `1.001` -> `2.001`.
* JS evaluates at `1` -> `NaN` (division by zero).
* Result: Limit is approx 2.
* Chart draws curve, shows asymptote/point at x=1.
* Table shows values.
* Looks good.

* User types `sin(x)/x`.
* User types `0`.
* JS evaluates at `0.001` -> `0.9999…`
* JS evaluates at `-0.001` -> `0.9999…`
* Result: Limit is approx 1.
* Chart draws sinc function.

* User types `hello`.
* Error message “Invalid function format” appears below input.

* User clicks “Copy”.
* Text copied to clipboard.

* User clicks “Reset”.
* Inputs cleared/defaulted.

* Mobile view.
* Table scrolls horizontally.
* Chart shrinks to fit.

Everything seems covered.

**One detail on “Limit Laws” in the article:**
I need to make sure the article explains *what* the laws are (Sum, Product, Quotient, Power) so the calculator context makes sense.

**Article Outline:**
H1: Calculating Limits Using the Limit Laws: The Ultimate Guide
H2: What Are Limit Laws?
H2: The Mathematical Foundation of Limits
H2: How to Use Our Limit Laws Calculator
H2: Practical Examples of Calculating Limits
H2: Key Factors That Influence Limit Calculations
H2: Frequently Asked Questions (FAQ)
H2: Related Calculus Tools and Resources

This looks solid.

**Code Generation:**
I will now generate the HTML string.

**Note on `var` usage in JS:**
I will declare all variables with `var`.
I will use `function functionName() { … }` syntax.
I will use `if (…) { … } else { … }`.
I will use `for (var i = 0; … ) { … }`.

**Note on Chart:**
I will use a simple line chart on HTML5 Canvas.
X-axis: `a – 2` to `a + 2`.
Y-axis: Auto-scale based on min/max Y in that range.

**Note on Table:**
Standard HTML table.

**Note on Copy:**
`navigator.clipboard.writeText` might fail in some insecure contexts (http), but usually works in https/file. I’ll add a fallback `window.clipboardData` for IE/older browsers just in case, or just stick to the modern API as it’s standard now. The prompt asks for “professional”, so modern API is fine.

**Note on “Limit Laws” Calculator Name:**
I’ll call it “Limit Laws Calculator” to match the keyword.

**Note on “Input section”:**
The prompt says “Main wrapper class: `.date-calc-container`”. I will use this exact class.

**Note on “Input IDs”:**
`funcInput`, `limitPoint`.

**Note on “Output IDs”:**
`mainResult`, `leftResult`, `rightResult`, `evalResult`.

**Note on “Helper Text”:**
“Enter a mathematical function f(x)…”

**Note on “Error messages”:**
`

**Note on “Reset”:**
``

**Note on “Copy”:**
``

**Note on “Real-time”:**
``
``

**Note on “Chart”:**
``

**Note on “Table”:**
`

…

**Note on “Responsive”:**
``

**Note on “Colors”:**
`#004a99`, `#28a745`, `#f8f9fa`.

**Note on “SEO”:**
Title, Meta Description, H1, Content.

**Note on “Internal Links”:**
I will insert them in the text.

**Note on “Keyword Density”:**
I will ensure the phrase “calculating limits using the limit laws” appears often.

**Note on “Examples”:**
Example 1: Polynomial.
Example 2: Rational function (0/0 form).

**Note on “FAQ”:**
8 questions.

**Note on “Related Tools”:**
6 links.

**Note on “Single Column”:**
CSS: `.container { max-width: 1000px; margin: 0 auto; padding: 20px; }`

**Note on “No Markdown”:**
I will output the code block directly.

**Note on “No Explanations”:**
I will provide only the HTML code.

**Note on “DO NOT use placeholders like ‘// logic here'”:**
I will write the full logic.

**Note on “Ensure your document.getElementById calls MATCH EXACTLY”:**
I will double check.

**Note on “Handle edge cases”:**
If `a` is infinity? I’ll handle finite `a` for now as it’s a standard calculator. If `a` is very large, the chart might look weird, but I’ll clamp the chart range to `a +/- 2` or `a +/- a*0.1` if `a` is large.

**Note on “Input labels”:**
“Function f(x)”, “Approaching Value (a)”.

**Note on “Result”:**
“Limit Value”.

**Note on “Intermediate values”:**
“Left-Hand Limit”, “Right-Hand Limit”, “Direct Substitution”.

**Note on “Explanation of formula”:**
“The calculator evaluates the function at points infinitesimally close to a…”

**Note on “Chart Series”:**
Series 1: Function Curve.
Series 2: Limit Point (visualized as a distinct dot).

**Note on “Mobile Table”:**
`

…

` with `overflow-x: auto`.

**Note on “Mobile Chart”:**
CSS `canvas { max-width: 100%; }`.

**Note on “Inline onclick”:**
Yes.

**Note on “var only”:**
Yes.

**Note on “No markdown fences”:**
Yes.

**Note on “Output ONLY complete, valid HTML code”:**
Yes.

**Note on “Topic: calculating limits using the limit laws”:**
Yes.

**Note on “NOT a loan calculator”:**
Yes.

**Note on “REMOVE all ‘$’ signs”:**
Yes.

**Note on “RENAME inputs”:**
Yes.

**Note on “Example: If topic is ‘BMI Calculator’…”:**
Yes.

**Note on “Professional Date Visual Style”:**
Yes.

**Note on “Responsive Design for Mobile”:**
Yes.

**Note on “Tables & Charts (MANDATORY)”:**
Yes.

**Note on “SEO LONG-FORM ARTICLE REQUIREMENTS”:**
Yes.

**Note on “KEYWORD DENSITY & INTERNAL LINKING (CRITICAL)”:**
Yes.

**Note on “SEO & STRUCTURE REQUIREMENTS”:**
Yes.

**Note on “DELIVERY FORMAT (STRICT)”:**
Yes.

**Note on “VARIABLE PLACEHOLDERS”:**
I will replace them in the content.

**Final check on “Limit Laws” specific calculator logic:**
Since I can’t do symbolic math, I will focus on the **Direct Substitution Property** and **Numerical Estimation**.
The calculator will:
1. Parse $f(x)$.
2. Evaluate $f(a – 0.0001)$ and $f(a + 0.0001)$.
3. Average them or check if they are close.
4. Display the limit.
5. Explain that this demonstrates the limit laws (specifically the definition of a limit).

This satisfies the prompt’s requirement for a calculator on this topic.

**Let’s write the code.**

**HTML Head:**
Title: Limit Laws Calculator & Guide | Calculate Limits Online
Meta Description: Master calculating limits using the limit laws with our free online calculator. Includes step-by-step examples, formulas, and graphing.

**HTML Body:**
Header: Limit Laws Calculator
Subheader: Evaluate limits and visualize the approaching behavior.

**Calculator Inputs:**
1. Function f(x): `(x^2 – 1) / (x – 1)`
2. Approach value (a): `1`

**Calculator Outputs:**
1. Limit: `2`
2. f(a – 0.001): `1.999`
3. f(a + 0.001): `2.001`
4. f(a): `NaN` (if undefined)

**Chart:**
Canvas showing the “hole” at x=1 and the curve approaching y=2.

**Article:**
Deep content about limit laws.

**Internal Links:**
/tools/derivative-calculator
/tools/integral-calculator
/learn/calculus-basics
/learn/continuity
/tools/scientific-calculator
/blog/math-tutorials

**Keywords:**
calculating limits using the limit laws
limit laws
limit calculator
evaluate limits
direct substitution

**Ready.**

**One final check on JS:**
`var` is used.
`document.getElementById` is used correctly.
Logic is complete.
No external libs.

**Chart implementation details:**
I need to map data values to canvas coordinates.
`map(val, min, max, canvasMin, canvasMax)`.
Draw axes.
Draw grid.
Draw curve.
Draw limit point.

**Table implementation details:**
Simple HTML string construction.

**Copy implementation:**
`var text = “Limit: ” + result + …`
`navigator.clipboard.writeText(text)`

**Reset implementation:**
`document.getElementById(“funcInput”).value = “(x^2 – 4)/(x – 2)”;`
`document.getElementById(“limitPoint”).value = “2”;`
`calculate();`

**Error handling:**
`try { eval(func) } catch (e) { show error }`

**Helper text:**
“Use x as variable. Ex: 2*x + 1, sin(x), x^2.”

**Let’s generate.**

**One detail on “Limit Laws” in the article:**
I will explicitly list the laws:
1. Sum Law
2. Difference Law
3. Product Law
4. Quotient Law
5. Power Law
6. Constant Law

And explain how the calculator helps verify these.

**One detail on “Keyword Density”:**
I will write a long article (~2000 words) to ensure natural density.

**One detail on “Input Validation”:**
If user types garbage, show “Please enter a valid function”.

**One detail on “Chart Update”:**
Call `drawChart()` inside `calculate()`.

**One detail on “Table Update”:**
Call `updateTable()` inside `calculate()`.

**One detail on “Responsive Chart”:**
The canvas will have `width=”600″` and `height=”300″` in HTML attributes, but CSS `max-width: 100%`. The JS will need to handle scaling or just let CSS handle the visual scaling (which is usually fine for simple line charts). I’ll stick to CSS scaling for simplicity and robustness.

**One detail on “Table Scroll”:**
`

**One detail on “Colors”:**
Primary: #004a99.
Success: #28a745.
Background: #f8f9fa.
Text: #333.

**One detail on “Fonts”:**
font-family: -apple-system, BlinkMacSystemFont, “Segoe UI”, Roboto, “Helvetica Neue”, Arial, sans-serif;

**One detail on “Layout”:**
`.date-calc-container` will have `background: white`, `padding: 30px`, `border-radius: 8px`, `box-shadow: 0 2px 10px rgba(0,0,0,0.1)`.

**One detail on “Inputs”:**
Styled with `padding: 10px`, `border: 1px solid #ccc`, `border-radius: 4px`.

**One detail on “Buttons”:**
Styled with `background: #004a99`, `color: white`, `padding: 10px 20px`, `border: none`, `border-radius: 4px`, `cursor: pointer`.

**One detail on “Result Box”:**
`background: #f0f7ff`, `border-left: 5px solid #004a99`.

**One detail on “Chart Legend”:**
“Function f(x)” and “Limit Point”.

**One detail on “Chart Caption”:**
“Figure 1: Graphical representation of the function near the limit point.”

**One detail on “Table Caption”:**
“Table 1: Numerical verification of the limit.”

**One detail on “Article Section Order”:**
A, B, C, D, E, F, G.

**One detail on “Internal Links in G”:**
List format.

**One detail on “FAQ”:**
Accordion or just Q&A blocks? Just Q&A blocks is simpler and better for SEO/scannability.

**One detail on “Primary Keyword in H1”:**
“Limit Laws Calculator”.

**One detail on “Primary Keyword in Title”:**
“Limit Laws Calculator”.

**One detail on “Primary Keyword in Meta”:**
“Limit Laws Calculator”.

**One detail on “Primary Keyword in Summary”:**
“Limit Laws Calculator”.

**One detail on “Primary Keyword in Article”:**
“calculating limits using the limit laws”.

**One detail on “Related Keywords”:**
“limit laws”, “evaluate limits”, “direct substitution”.

**One detail on “Internal Links”:**
Anchor texts: “Derivative Calculator”, “Integral Calculator”, “Calculus Basics”, “Continuity Theorem”, “Scientific Calculator”, “Math Tutorials”.

**One detail on “Examples”:**
1. Polynomial: $\lim_{x \to 2} (3x^2 + 5)$
2. Rational: $\lim_{x \to 1} \frac{x^2 – 1}{x – 1}$

**One detail on “Key Factors”:**
1. Continuity
2. Domain

`…`

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