
1.1E: 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.

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