#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

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#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.

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

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

From the table, we can see that $f(x)$ gets nearer and nearer to 0.5 as $x$ 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.

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

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

From the graph, we can see that $f(x)$ gets nearer and nearer to 0.5 as $x$ 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.

#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 $x$ becomes unbounded in the positive or negative direction.

#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.

#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 $x$ gets closer and closer to the $x$-value of the vertical line.

#How can I identify vertical asymptotes using limits?

A function will have a vertical asymptote at any $x$-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.

### Worked Example Let $f$ 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 $f$.

Answer:

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

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

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

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

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

$\frac{5}{x - 2}$ becomes closer and closer to zero as $x$ 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 $f$ has a horizontal asymptote at $y = 3$.

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

#Glossary

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

#Practice Questions

Practice Question

1. Given the function $f(x) = \frac{2x^2 - 3x}{x^2 - 1}$, determine the horizontal asymptote.

Practice Question

2. For the function $g(x) = \frac{1}{x - 3}$, identify the vertical asymptote.

Practice Question

3. Estimate the limit of the function $h(x) = \frac{\sin x}{x}$ as $x$ 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 $x$ 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.