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Limits

Sarah Miller

Sarah Miller

6 min read

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Study Guide Overview

This study guide covers infinite limits and limits at infinity. It explains the concepts of infinite limits and their connection to vertical asymptotes. It also discusses limits at infinity and their relationship with horizontal asymptotes. The guide includes worked examples, practice questions, and a glossary of key terms like vertical asymptote and horizontal asymptote.

Study Notes on Infinite Limits and Limits at Infinity

Table of Contents

  1. Infinite Limits
  2. Limits at Infinity
  3. Practice Questions
  4. Glossary
  5. Summary and Key Takeaways

Infinite Limits

What is an Infinite Limit?

Infinite limits occur when the values of a function become unbounded (either positively or negatively) as xx approaches a specific value.

Key Concept

If the value of a function ff increases without bound as xx approaches some value cc, we write: lim⁡x→cf(x)=∞\lim_{{x \to c}} f(x) = \infty

If the value of a function ff decreases without bound as xx approaches some value cc, we write: lim⁡x→cf(x)=−∞\lim_{{x \to c}} f(x) = -\infty

For example: lim⁡_x→01x2=∞\lim\_{{x \to 0}} \frac{1}{x^2} = \infty lim⁡_x→0(−1x2)=−∞\lim\_{{x \to 0}} \left(-\frac{1}{x^2}\right) = -\infty One-sided limits can be different. For example: lim⁡_x→0+1x=∞\lim\_{{x \to 0^+}} \frac{1}{x} = \infty lim⁡_x→0−1x=−∞\lim\_{{x \to 0^-}} \frac{1}{x} = -\infty

Connection with Vertical Asymptotes

When a function has an infinite limit at a point, its graph has a vertical asymptote at that value of xx.

Key Concept

A vertical asymptote is a vertical line that the graph approaches but never touches or intersects as xx approaches a certain value.

Identifying infinite limits helps in identifying the location of vertical asymptotes on the graph of the function.

Worked Example

Consider the function ff defined by: f(x)=1(x−2)2f(x) = \frac{1}{(x-2)^2}

This function has a vertical asymptote at x=2x = 2.

To find the limit as xx approaches 2: lim⁡x→2f(x)=∞\lim_{{x \to 2}} f(x) = \infty

Since the denominator approaches zero as xx approaches 2, the value of the function increases without bound.

Limits at Infinity

What is a Limit at Infinity?

Limits at infinity describe the behavior of a function as xx increases or decreases without bound.

Key Concept

When considering the behavior of a function ff as xx increases without bound, we write: lim⁡x→∞f(x)\lim_{{x \to \infty}} f(x)

When considering the behavior of a function ff as xx decreases without bound, we write: lim⁡x→−∞f(x)\lim_{{x \to -\infty}} f(x)

For example: lim⁡_x→∞(x+1)=∞\lim\_{{x \to \infty}} (x + 1) = \infty lim⁡_x→−∞(x+1)=−∞\lim\_{{x \to -\infty}} (x + 1) = -\infty

But for other functions, their values approach a fixed value: lim⁡x→∞(1x+1)=1\lim_{{x \to \infty}} \left( \frac{1}{x} + 1 \right) = 1 lim⁡x→−∞(1x+1)=1\lim_{{x \to -\infty}} \left( \frac{1}{x} + 1 \right) = 1

Connection with Horizontal Asymptotes

When a function has a finite limit at infinity, its graph has a horizontal asymptote at that value of yy.

Key Concept

A horizontal asymptote is a horizontal line that the graph approaches but generally never touches or intersects as xx becomes unbounded.

Identifying finite limits at infinity helps in identifying the location of horizontal asymptotes on the graph of the function.

Exam Tip

Exam Tip

If an exam question uses a function to model a real-world scenario, be sure to interpret any limits at infinity in the context of the question.

Worked Example

Consider the function PP defined by: P(t)=4−3t+1,t≥0P(t) = 4 - \frac{3}{t+1}, \quad t \geq 0

This function is used to model the population (in hundreds) of squirrels in a woodland area tt years after the beginning of a study.

(a) Find: lim⁡t→∞P(t)\lim_{{t \to \infty}} P(t)

Answer: As tt increases, 3t+1\frac{3}{t+1} approaches zero, so: lim⁡t→∞P(t)=4\lim_{{t \to \infty}} P(t) = 4

(b) Interpret your answer in the context of the problem.

Answer: Over time, the population of squirrels in the woodland will approach 400. ## Practice Questions

Practice Question
  1. Evaluate: lim⁡x→−3+1x+3\lim_{{x \to -3^+}} \frac{1}{x+3}
  2. Determine the vertical asymptotes of the function g(x)=2xx2−4g(x) = \frac{2x}{x^2 - 4}.
  3. Find: lim⁡x→∞(2x+1x)\lim_{{x \to \infty}} \left( \frac{2x+1}{x} \right)
  4. Explain the behavior of h(x)=3x2−x+1x2+1h(x) = \frac{3x^2 - x + 1}{x^2 + 1} as x→−∞x \to -\infty.

Glossary

  • Infinite Limit: A limit in which the function values become unbounded as xx approaches a certain value.
  • Vertical Asymptote: A vertical line that a graph approaches but never touches or intersects as xx approaches a certain value.
  • Limit at Infinity: The behavior of a function as xx increases or decreases without bound.
  • Horizontal Asymptote: A horizontal line that a graph approaches but generally never touches or intersects as xx becomes unbounded.

Summary and Key Takeaways

  • Infinite Limits occur when the function values become unbounded as xx approaches a specific value.
  • Vertical Asymptotes indicate the presence of infinite limits at certain values of xx.
  • Limits at Infinity describe the behavior of a function as xx increases or decreases without bound.
  • Horizontal Asymptotes indicate finite limits at infinity.

Remember to always connect your mathematical findings with the context of the problem, especially in real-world scenarios. This will help you interpret the results accurately and provide meaningful conclusions.

Previous Topic - Definition of a LimitNext Topic - Properties of Limits

Question 1 of 12

What does the notation lim⁡x→cf(x)=∞\lim_{x \to c} f(x) = \infty signify about the function f(x)f(x) as xx approaches cc?

The function approaches zero

The function decreases without bound

The function increases without bound

The function approaches a finite value