2.1E: Exercises

Exercise $\PageIndex{1}$

Terms and Concepts

1. In your own words, what does it mean to "find the limit of $f(x)$ as $x$ approaches 3"?
2. An expression of the form $\frac{0}{0}$ is called _____.
3. T/F: The limit of $f(x)$ as $x$ approaches 5 is $f(5)$.
4. 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

$\lim_{x \to \infty}\left(1+\dfrac{1}{x}\right)^x=2.7183=e.$