Math Practice 1

Write your name on this paper. Explain all your answers. Use common sense . Do
not use calculators /book/notes. There are six parts all equally weighted. Here is the first
one:

1. Find the following limits and explain which limit laws you used .
(a)

(b) where f is the function defined by the rule

2. Miscellaneous questions:

(a) What is ?
(b) Does the function f(x) = x² have an inverse? If so, what is it? If not, why not?
(c) What is the domain of the function

3. Evaluate the following limits.
(a)

(b)

(c)

4. The function

is differentiable whenever x does not equal 1. Compute the derivative of f at the point
a, a ≠ 1. (Use only the material from Chapters 1 and 2, i.e., use the definition of the
derivative we just discussed in the lectures.)

5. Give correct definitions of the following notions.

(a) Define the inverse of a function f.
(b) Define what it means for f to be increasing on the interval I.
(c) Define what it means for f to be differentiable on the interval (a, b).

6. Continuity, etc.
(a) Find a > 0 such that for all x with 0 < |x − 1| < we have

(b) Find an integer n such that f(x) = x³ − 5x² − 4x − 5 has a zero between n and
n + 1. As always, explain why your answer is true.