#Study Notes on Infinite Limits and Limits at Infinity

#Table of Contents

Infinite Limits
Limits at Infinity
Practice Questions
Glossary
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 $x$ approaches a specific value.

Key Concept

If the value of a function $f$ increases without bound as $x$ approaches some value $c$ , we write: $\lim_{{x \to c}} f(x) = \infty$

If the value of a function $f$ decreases without bound as $x$ approaches some value $c$ , we write: $\lim_{{x \to c}} f(x) = -\infty$

For example:

\lim\_{{x \to 0}} \frac{1}{x^2} = \infty

\lim\_{{x \to 0}} \left(-\frac{1}{x^2}\right) = -\infty

One-sided limits can be different. For example:

\lim\_{{x \to 0^+}} \frac{1}{x} = \infty

\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 $x$ .

Key Concept

A vertical asymptote is a vertical line that the graph approaches but never touches or intersects as $x$ 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 $f$ defined by: $f(x) = \frac{1}{(x-2)^2}$

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

To find the limit as $x$ approaches 2: $\lim_{{x \to 2}} f(x) = \infty$

Since the denominator approaches zero as $x$ 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 $x$ increases or decreases without bound.

Key Concept

When considering the behavior of a function $f$ as $x$ increases without bound, we write: $\lim_{{x \to \infty}} f(x)$

When considering the behavior of a function $f$ as $x$ decreases without bound, we write: $\lim_{{x \to -\infty}} f(x)$

For example:

\lim\_{{x \to \infty}} (x + 1) = \infty

\lim\_{{x \to -\infty}} (x + 1) = -\infty

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

#Connection with Horizontal Asymptotes

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

Key Concept

A horizontal asymptote is a horizontal line that the graph approaches but generally never touches or intersects as $x$ 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 $P$ defined by: $P(t) = 4 - \frac{3}{t+1}, \quad t \geq 0$

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

(a) Find: $\lim_{{t \to \infty}} P(t)$

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

(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

Evaluate: $\lim_{{x \to -3^+}} \frac{1}{x+3}$
Determine the vertical asymptotes of the function $g(x) = \frac{2x}{x^2 - 4}$ .
Find: $\lim_{{x \to \infty}} \left( \frac{2x+1}{x} \right)$
Explain the behavior of $h(x) = \frac{3x^2 - x + 1}{x^2 + 1}$ as $x \to -\infty$ .

#Glossary

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

#Summary and Key Takeaways

Infinite Limits occur when the function values become unbounded as $x$ approaches a specific value.
Vertical Asymptotes indicate the presence of infinite limits at certain values of $x$ .
Limits at Infinity describe the behavior of a function as $x$ 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.