1.1E: Exercises
- Page ID
- 11018
This page is a draft and is under active development.
Exercise \(\PageIndex{1}\)
Terms and Concepts
- In your own words, what does it mean to "find the limit of \(f(x)\) as \(x\) approaches 3"?
- An expression of the form \(\frac{0}{0}\) is called _____.
- T/F: The limit of \(f(x)\) as \(x\) approaches 5 is \(f(5)\).
- Describe three situations where \(\lim\limits_{x\to c}f(x)\) does not exist.
Exercise \(\PageIndex{2}\)
For exercises 1 - 2, consider the function \(f(x)=\dfrac{x^2−1}{|x−1|}\).
1) [T] Complete the following table for the function. Round your solutions to four decimal places.
\(x\) | \(f(x)\) | \(x\) | \(f(x)\) |
---|---|---|---|
0.9 | a. | 1.1 | e. |
0.99 | b. | 1.01 | f. |
0.999 | c. | 1.001 | g. |
0.9999 | d. | 1.0001 | h. |
2) What do your results in the preceding exercise indicate about the two-sided limit \(\displaystyle \lim_{x→1}f(x)\)? Explain your response.
- Answer
-
\(\displaystyle \lim_{x \to 1}f(x)\) does not exist because \(\displaystyle \lim_{x \to 1^−}f(x)=−2≠\lim_{x \to 1^+}f(x)=2\).
Exercise \(\PageIndex{3}\)
Consider the function \(f(x)=\left(1+\dfrac{1}{x}\right)^x\). Make a table showing \(f(x)\) for \(x=1,2,3, .....\). Round your solutions to five decimal places. What can you say about the value of the function \(f(x)\) as \(x\) increases indefinitely?
- Answer
-
\(\lim_{x \to \infty}\left(1+\dfrac{1}{x}\right)^x=2.7183=e.\)