Limits

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
- Limits from Tables
- Limits from Graphs
- Horizontal Asymptotes
- Vertical Asymptotes
- Glossary
- Practice Questions
- 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.
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 defined by: [ f(x) = \frac{1 - \cos x}{x^2} ]
The table below shows values of near . Note that the function is not defined at because .
-0.1 | 0.49958347 |
-0.01 | 0.49999583 |
-0.001 | 0.49999996 |
0 | not defined |
0.001 | 0.49999996 |
0.01 | 0.49999583 |
0.1 | 0.49958347 |
From the table, we can see that gets nearer and nearer to 0.5 as gets closer to 0.
#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.
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 defined by: [ f(x) = \frac{1 - \cos x}{x^2} ]
The graph shows the behavior of near . Note that the function is not defined at because .
From the graph, we can see that gets nearer and nearer to 0.5 as approaches 0.
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 becomes unbounded in the positive or negative direction.
A horizontal asymptote is a horizontal line that the graph of a function gets closer to as goes to or .
#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.
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 gets closer and closer to the -value of the vertical line.
A vertical asymptote is a vertical line that the graph of a function gets closer to as approaches a certain value.
#How can I identify vertical asymptotes using limits?
A function will have a vertical asymptote at any -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.
By graphing a function on your graphing calculator, you can spot any unbounded behavior by a function at certain values of and check that vertical asymptotes you determine analytically are correct.
Using limits, identify the vertical and horizontal asymptotes (if any) on the graph of .
Answer:
The denominator becomes 0 when , so start by considering the limits there.
At , the numerator is equal to -5, so zero only occurs in the denominator. Just 'to the left' of 2, and so: [ \lim_{{x \to 2^-}} f(x) = \infty ]
Just 'to the right' of 2, and so: [ \lim_{{x \to 2^+}} f(x) = -\infty ]
This confirms that the graph of has a vertical asymptote at .
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} ]
becomes closer and closer to zero as 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 has a horizontal asymptote at .
Conclusion: The graph of has a vertical asymptote at and a horizontal asymptote at .
#Glossary
- Limit: The value that a function approaches as the input approaches some value.
- Horizontal Asymptote: A horizontal line that a graph approaches as goes to infinity or negative infinity.
- Vertical Asymptote: A vertical line that a graph approaches as approaches a specific value, indicating the function becomes unbounded.
#Practice Questions
Practice Question
- Given the function , determine the horizontal asymptote.
Practice Question
- For the function , identify the vertical asymptote.
Practice Question
- Estimate the limit of the function as 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 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.
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