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title: Worksheet on definition of derivative and limits

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# Worksheet on definition of derivative and limits

1. Use the definition of the derivative to find $f'(3)$ when $f(x) = \sqrt{x}.$

2. Use the definition of the derivative to find $f'(-1)$ when $f(x) = 2x-x^2$.

3. Find the limit $$\lim_{x\to0}\frac{\cos(x)-1}{x}.$$

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# Practice questions for Exam 1

1. (Section 2.3, \#96) Find the limit $$\lim_{x\to0}\frac{(3+x)^2-9}{x}$$
2. (Section 2.3, \#99) Find the limit $$\lim_{\theta\to\pi}\frac{\sin(\theta)}{\tan(\theta)}$$
3. (Section 3.1, \#19) Use the definition of the derivative to compute $f'(-4)$ when $f(x) = \frac{2}{x+3}$.
4. (Section 2.4, \#139) Decide if the function is continuous at the given point. If discontinuous, what type of discontinuity is it? $$f(x)=\frac{2x^2-5x+3}{x-1}\ \text{at }x=1.$$
5. (Section 4.6, \#269) Find the horizontal asymptote of the function $$h(x) = \frac{4x}{\sqrt{x^2-1}}.$$
6. (Section 2.2, \#47--49) The graph of a function $f(x)$ is shown in the figure below. Which of the following statements are True and which are False? If a statement is false, explain why.
  a. $$\lim_{x\to-2^+}f(x) = 3,$$
  b. $$\lim_{x\to-8}f(x) = f(-8),$$
  c. $$\lim_{x\to6}f(x) = 5.$$
  
![](https://i.imgur.com/YsBCOxJ.jpg)
