# MATH 2301-102 and -103 Fall 2017 Calculus I Recitation 3 Week 3

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

1. For $$f(x)=x^3$$, compute $$\displaystyle \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=$$
2. For $$f(x)= (x+1)^{-1}$$, compute $$\displaystyle \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=$$
3. Compute $$\displaystyle \lim_{z\rightarrow x^{-}} \frac{z^2-x^2}{|x-z|}=$$
1. Sketch the graph of a single function that is continuous except:
1. at 1, where it has a removable discontinuity;
2. at 2, where it has a jump discontinuity but is continuous from the right; and
3. at 3, where it has an infinite discontinuity.
2. Let $f(x)= \begin{cases} {\displaystyle\frac{2x^2-x-15}{x-3}} & \text{if $$x \lt 3$$}\\ kx-1 & \text{if $$x \ge 3$$} \end{cases}\, .$ Determine the value of $$k$$ that will make the function $$f$$ continuous at 3, or explain why no value of $$k$$ will work.
1. State the Intermediate Value Theorem. Identify what are its assumptions (hypotheses) and what are its conclusions.
2. Use the Intermediate Value Theorem to show that the equation $$x^2 =\cos(x)$$ has a solution.