# MATH 2301-102 and -103 Fall 2017 Calculus I Recitation 4 Week 4

See the coversheet for instructions and the point value of each problem.

1. Sketch the graph of a function $$f$$ for which:
1. It has a removable discontinuity at $$x=1$$,
2. $$\lim_{x\rightarrow 2^-}f(x)=+\infty$$,
3. $$\lim_{x\rightarrow 2^+}f(x)=-1$$,
4. $$\lim_{x\rightarrow +\infty}f(x)=3$$,
5. $$f'(x)>0$$ everywhere it exists.
2. Let $$f(x)=\sqrt{x}-2$$.
1. State the definition of the derivative as a limit.
2. Using this definition, compute $$f'(x)$$.
3. Find the equation for the tangent line at $$x=9$$.
4. Graph $$f(x)$$ and the tangent line.
3. Find values for $$m$$ and $$b$$ so that $$\displaystyle f(x)= \begin{cases} x^2 & \text{if \(x\le -2$$}\\ mx+b & \text{if $$x> -2$$} \end{cases}\) is differentiable at $$x=-2$$.
4. Compute the following limits. If you use the squeeze theorem, then indicate the two functions that you are using to squeeze.
1. $$\displaystyle\lim_{x\rightarrow \infty}x\sin(x^{-1})$$
2. $$\displaystyle\lim_{x\rightarrow \infty}\frac{\sin(x)}{x}$$
3. $$\displaystyle\lim_{x\rightarrow \infty}x\cos(x^{-1})$$
4. $$\displaystyle\lim_{x\rightarrow \infty} (x-x^2)$$
5. $$\displaystyle\lim_{x\rightarrow \infty} (x-\sin(x))$$