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Limits

Sarah Miller

Sarah Miller

7 min read

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Study Guide Overview

This study guide covers evaluating limits both numerically (using tables) and graphically. It explains how to estimate limits from tables and graphs, and how to identify horizontal and vertical asymptotes using limits. The guide also includes a glossary, practice questions, and key takeaways summarizing the core concepts of limit evaluation and asymptote identification.

Study Notes on Limits and Asymptotes

Table of Contents

  1. Limits from Tables
  2. Limits from Graphs
  3. Horizontal Asymptotes
  4. Vertical Asymptotes
  5. Glossary
  6. Practice Questions
  7. Summary and Key Takeaways

Limits from Tables

How can I estimate a limit using values in a table?

A table can show the behavior of a function near a point, allowing you to estimate the limit at that point.

Key Concept

Values of a function in a table can show the behavior of a function near a point, which can allow you to estimate the limit at that point.

For example, consider the function ff defined by: [ f(x) = \frac{1 - \cos x}{x^2} ]

The table below shows values of f(x)f(x) near x=0x = 0. Note that the function is not defined at x=0x = 0 because f(0)=00f(0) = \frac{0}{0}.

xxf(x)f(x)
-0.10.49958347
-0.010.49999583
-0.0010.49999996
0not defined
0.0010.49999996
0.010.49999583
0.10.49958347

From the table, we can see that f(x)f(x) gets nearer and nearer to 0.5 as xx gets closer to 0. Therefore, we can estimate that: [ \lim_{{x \to 0}} f(x) = 0.5 ] However, analytical methods would need to be used to confirm that this is indeed the limit.

Limits from Graphs

How can I estimate a limit using a graph?

A graph can show the behavior of a function near a point, allowing you to estimate the limit at that point.

Key Concept

Graphs can show the behavior of a function near a point, which can allow you to estimate the limit at that point.

For example, consider the function ff defined by: [ f(x) = \frac{1 - \cos x}{x^2} ]

The graph shows the behavior of f(x)f(x) near x=0x = 0. Note that the function is not defined at x=0x = 0 because f(0)=00f(0) = \frac{0}{0}.

From the graph, we can see that f(x)f(x) gets nearer and nearer to 0.5 as xx approaches 0. Therefore, we can estimate that: [ \lim_{{x \to 0}} f(x) = 0.5 ] However, analytical methods would need to be used to confirm that this is indeed the limit.

Exam Tip

You can graph functions on your graphing calculator to check your answers when determining limits analytically.

Horizontal Asymptotes

What is a horizontal asymptote?

A horizontal asymptote is a horizontal line that the graph of a function gets closer and closer to (but never touches or intersects) as xx becomes unbounded in the positive or negative direction.

Key Concept

A horizontal asymptote is a horizontal line that the graph of a function gets closer to as xx goes to \infty or -\infty.

How can I identify horizontal asymptotes using limits?

A function will have a horizontal asymptote if it has a finite limit at infinity:

[ \lim_{{x \to \infty}} f(x) = c ] or [ \lim_{{x \to -\infty}} f(x) = c ]

Horizontal asymptotes (if any) can be determined by evaluating the limits at infinity.

Exam Tip

By graphing a function on your graphing calculator, you can spot any asymptotic behavior by a function at plus or minus infinity and check limits that you have determined analytically.

Vertical Asymptotes

What is a vertical asymptote?

A vertical asymptote is a vertical line that the graph of a function gets closer and closer to (but never touches or intersects) as xx gets closer and closer to the xx-value of the vertical line.

Key Concept

A vertical asymptote is a vertical line that the graph of a function gets closer to as xx approaches a certain value.

How can I identify vertical asymptotes using limits?

A function will have a vertical asymptote at any xx-value where the function becomes unbounded:

[ \lim_{{x \to c^-}} f(x) = \pm \infty ] or [ \lim_{{x \to c^+}} f(x) = \pm \infty ]

Vertical asymptotes (if any) can be determined by identifying points where the function becomes unbounded. Usually, this will involve a function in the form of a quotient at points where the denominator becomes zero.

Exam Tip

By graphing a function on your graphing calculator, you can spot any unbounded behavior by a function at certain values of xx and check that vertical asymptotes you determine analytically are correct.

### Worked Example Let ff be the function defined by: \[ f(x) = \frac{3x - 11}{x - 2} \]

Using limits, identify the vertical and horizontal asymptotes (if any) on the graph of ff.

Answer:

The denominator becomes 0 when x=2x = 2, so start by considering the limits there.

At x=2x = 2, the numerator is equal to -5, so zero only occurs in the denominator. Just 'to the left' of 2, 3x11<03x - 11 < 0 and x2<0x - 2 < 0 so: [ \lim_{{x \to 2^-}} f(x) = \infty ]

Just 'to the right' of 2, 3x11<03x - 11 < 0 and x2>0x - 2 > 0 so: [ \lim_{{x \to 2^+}} f(x) = -\infty ]

This confirms that the graph of ff has a vertical asymptote at x=2x = 2.

To identify horizontal asymptotes, start by rearranging to make the behavior of the function more obvious: [ \frac{3x - 11}{x - 2} = \frac{3(x - 2) - 5}{x - 2} = \frac{3(x - 2)}{x - 2} - \frac{5}{x - 2} = 3 - \frac{5}{x - 2} ]

5x2\frac{5}{x - 2} becomes closer and closer to zero as xx increases in the positive or negative directions, so: [ \lim_{{x \to -\infty}} f(x) = \lim_{{x \to \infty}} f(x) = 3 - 0 = 3 ]

This means that the graph of ff has a horizontal asymptote at y=3y = 3.

Conclusion: The graph of ff has a vertical asymptote at x=2x = 2 and a horizontal asymptote at y=3y = 3.

Glossary

  • Limit: The value that a function approaches as the input approaches some value.
  • Horizontal Asymptote: A horizontal line that a graph approaches as xx goes to infinity or negative infinity.
  • Vertical Asymptote: A vertical line that a graph approaches as xx approaches a specific value, indicating the function becomes unbounded.

Practice Questions

Practice Question
  1. Given the function f(x)=2x23xx21f(x) = \frac{2x^2 - 3x}{x^2 - 1}, determine the horizontal asymptote.
Practice Question
  1. For the function g(x)=1x3g(x) = \frac{1}{x - 3}, identify the vertical asymptote.
Practice Question
  1. Estimate the limit of the function h(x)=sinxxh(x) = \frac{\sin x}{x} as xx approaches 0 using a table of values.

Summary and Key Takeaways

  • Estimating Limits: Use tables and graphs to estimate the limit of a function as it approaches a certain point.
  • Horizontal Asymptotes: Identify horizontal asymptotes by evaluating the limit of the function as xx approaches infinity or negative infinity.
  • Vertical Asymptotes: Identify vertical asymptotes by determining where the function becomes unbounded, usually where the denominator of a quotient becomes zero.

Remember to cross-check your analytical answers with graphical representations when possible, and use your graphing calculator as a tool to verify your findings.

Question 1 of 8

Consider the table of values for a function f(x)f(x) as xx approaches 2. Based on the trend, what is the estimated limit of f(x)f(x) as xx approaches 2? 🧐

x1.91.991.99922.0012.012.1
f(x)3.83.983.998undef4.0024.024.2

3

4

Undefined

3.5