This document provides a math problem set involving limits, continuity, derivatives, tangent lines, and other calculus concepts. Students are asked to evaluate limits, discuss discontinuities of functions, find the domain of continuity of a piecewise function, derive equations of tangent and normal lines, take derivatives, find equations of tangent lines, and apply theorems like the Intermediate Value Theorem. The problem set is intended as a review for an upcoming exam.

2. a. limx->0sin34x8x3

3. b. limx->01-cos2πxx2

4. c. limx->-πtanxπ+x

5. d. limx->0x3cos43x3-x2

6. Discuss the discontinuities (on what point(s) are they continuous, and what is the type of discontinuity if they are not continuous at a point) of the following functions (5 pts each):

7. a. gx=1, x∈Z 0, otherwise

8. b. hxx+2, x<-21x, -2≤x≤2-x2+x+4, x>2

11. Suppose fx is a continuous function on [0,2], with f0=f2. Use IVT (or IZT) to show that for some c∈0,1, fc=fc+1. HINT: consider the function gx=fx+1-fx (10 pts)

12. Suppose fx≤M, ∀x∈R, where M is a positive constant. Furthermore, suppose that limx->agx=0. Use the squeeze theorem to find the value of

13. limx->afxgx