Limits & Continuity: Rules, Graphs & Business Uses

TAKEAWAYS:

• Limits help us understand what happens near a point, even if the function isn’t defined there.
• A function is continuous if its graph has no breaks, jumps, or holes—critical for stable business models.
• Left-hand and right-hand limits must match for continuity at a point.
• Limits and continuity help analyze trends in sales, costs, and production over time.
• Graphs visually show where functions break or jump, revealing potential business risks.
• The limit laws (sum, product, quotient) simplify complex calculations in economics.

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What is a Limit?

A limit tells us what value a function approaches as the input gets closer and closer to a certain point. It’s like asking: "What happens to the function’s output as we zoom in on a specific x-value?"

Key Idea: Approaching vs. Reaching

• Approaching: The function gets very close to a value, but may never actually reach it.
• Reaching: The function’s value equals the limit at that point.

Example: Simple Limit

Find ${\lim }_{x\to 2}(3x+1)$.

Step-by-Step Solution:

1. Direct Substitution: Plug $x=2$ into $3x+1$. $$3(2)+1=6+1=7$$
2. Conclusion: Since substitution works, the limit is 7.

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One-Sided Limits: Left and Right

Sometimes, a function behaves differently when approaching a point from the left or right. These are called one-sided limits.

Definitions:

• Left-hand limit (${\lim }_{x\to a^{-}}f(x)$): Approaches $a$ from values less than $a$.
• Right-hand limit (${\lim }_{x\to a^{+}}f(x)$): Approaches $a$ from values greater than $a$.

Example: Piecewise Function

Find ${\lim }_{x\to 3^{-}}f(x)$ and ${\lim }_{x\to 3^{+}}f(x)$ for: $$f(x)=\{\begin{array}{ll}{x}^2& \text{if }x<3\\ 2x+1& \text{if }x\ge 3\end{array}$$

Solution:

1. Left-hand limit ($x\to 3^{-}$): Use $f(x)=x^2$ because $x<3$. $$\underset{x\to 3^{-}}{\lim }f(x)=3^2=9$$
2. Right-hand limit ($x\to 3^{+}$): Use $f(x)=2x+1$ because $x\ge 3$. $$\underset{x\to 3^{+}}{\lim }f(x)=2(3)+1=7$$

Observation: The left and right limits are not equal (9 ≠ 7). This means the function has a jump discontinuity at $x=3$.

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Continuity: The Business Rule

A function is continuous at a point if:

1. The function is defined at that point.
2. The limit exists at that point.
3. The limit equals the function’s value at that point.

Types of Discontinuities:

Example: Checking Continuity

Is $f(x)=\frac{x^2-1}{x-1}$ continuous at $x=1$?

Steps:

1. Check if $f(1)$ is defined: $$f(1)=\frac{1-1}{1-1}=\frac{0}{0}\text{(Undefined!)}$$ → Not continuous because the function isn’t defined at $x=1$.

2. Find the limit (even though it’s undefined): Simplify $f(x)$: $$f(x)=\frac{(x-1)(x+1)}{x-1}=x+1\text{(for }x\ne 1\text{)}$$ Now, take the limit: $$\underset{x\to 1}{\lim }f(x)=\underset{x\to 1}{\lim }(x+1)=2$$ → The limit exists, but $f(1)$ doesn’t. This is a removable discontinuity.

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Limit Laws: Simplifying Calculations

Limit laws help break down complex functions into simpler parts. Here are the key ones:

Example: Applying Limit Laws

Find ${\lim }_{x\to 4}\frac{2x^2-5x-12}{x-4}$.

Steps:

1. Direct substitution gives $\frac{0}{0}$ (indeterminate). Factor the numerator: $$2x^2-5x-12=(2x+3)(x-4)$$
2. Simplify: $$\frac{(2x+3)(x-4)}{x-4}=2x+3\text{(for }x\ne 4\text{)}$$
3. Now take the limit: $$\underset{x\to 4}{\lim }(2x+3)=2(4)+3=11$$

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Business Applications: Why Limits Matter

Limits and continuity are used in business to:

1. Model Costs and Revenues:
   • Example: A factory’s cost function $C(x)$ might have a discontinuity at $x=100$ units (e.g., a sudden increase in labor costs).
2. Predict Trends:
   • If ${\lim }_{x\to \infty }P(x)=100$, it means profit $P(x)$ approaches $100 as production $x$ grows.
3. Avoid Sudden Changes:
   • A continuous profit function means no abrupt losses (e.g., no unexpected jumps in expenses).

Example: Profit Function

A company’s profit $P(x)$ (in thousands) is given by: $$P(x)=\{\begin{array}{ll}5x-10& \text{if }x<50\\ 0.5x^2-20x+1000& \text{if }x\ge 50\end{array}$$ Is $P(x)$ continuous at $x=50$?

Steps:

1. Check $P(50)$: $$P(50)=0.5(50{)}^2-20(50)+1000=1250-1000+1000=1250$$
2. Left-hand limit ($x\to 50^{-}$): $$\underset{x\to 50^{-}}{\lim }P(x)=5(50)-10=250-10=240$$
3. Right-hand limit ($x\to 50^{+}$): $$\underset{x\to 50^{+}}{\lim }P(x)=0.5(50{)}^2-20(50)+1000=1250$$
4. Conclusion:
   • Left limit (240) ≠ Right limit (1250).
   • Not continuous at $x=50$. This means there’s a sudden jump in profit at 50 units of production!

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Exam Tip: How to Score Full Marks

1. Always check for continuity in 3 steps:
   • Is the function defined at the point?
   • Does the limit exist?
   • Does the limit equal the function’s value?
2. For limits, try direct substitution first. If it fails (e.g., $\frac{0}{0}$), factor or simplify.
3. Graphs are your friends:
   • Sketch the function to visualize jumps, holes, or asymptotes.
   • Label left/right limits clearly.
4. Watch for common mistakes:
   • Forgetting to check both left and right limits for continuity.
   • Misapplying limit laws (e.g., quotient law when denominator is zero).
5. Business context:
   • If a question mentions "cost," "profit," or "revenue," think about real-world discontinuities (e.g., sudden tax hikes, bulk discounts).

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NEB-Style Practice Questions

Short Answer (5 marks each)

1. Find ${\lim }_{x\to 3}\frac{x^2-9}{x-3}$.
2. Determine if $f(x)=\{\begin{array}{ll}x+2& \text{if }x<1\\ 3x-1& \text{if }x\ge 1\end{array}$ is continuous at $x=1$.
3. A company’s revenue function is $R(x)=100x-0.5x^2$. Find ${\lim }_{x\to 200}R(x)$ and interpret it.

Long Answer (10 marks)

4. Consider the function: $$f(x)=\{\begin{array}{ll}\frac{\sin x}{x}& \text{if }x\ne 0\\ 1& \text{if }x=0\end{array}$$
   • Find ${\lim }_{x\to 0}f(x)$.
   • Is $f(x)$ continuous at $x=0$? Justify your answer.
   • Sketch the graph of $f(x)$ near $x=0$.

Multiple Choice (1 mark each)

5. Which of the following is not a type of discontinuity? a) Removable b) Jump c) Smooth d) Infinite
6. If ${\lim }_{x\to a}f(x)=L$ and ${\lim }_{x\to a}g(x)=M$, then ${\lim }_{x\to a}[f(x)+g(x)]$ is: a) $L+M$ b) $L\cdot M$ c) $\frac{L}{M}$ d) $L-M$

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Answers to Practice Questions:

1. 12 (Simplify to $x+3$).
2. No (Left limit = 3, Right limit = 2).
3. 10,000 (Revenue approaches $10,000 at 200 units).
4. a) 1 (Use ${\lim }_{x\to 0}\frac{\sin x}{x}=1$). b) Yes (Limit = $f(0)=1$).
5. c) Smooth.
6. a) $L+M$.