#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.