Name: First Midterm

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) lim_{x→2} \frac{3x^{2}+2x}{x^{2}+x}.

(b) lim_{x→10} f(x) where f is the function defined by the rule

f(x) = \{ \begin{matrix}2x − 10 & if & x > 10 \\ −1 & if & x = 10 \\ x & if & x < 10 \end{matrix}

2. Miscellaneous questions: (a) What is log_{1/3}(27)?

(b) Does the function f(x) = x^{2} have an inverse? If so, what is it? If not, why not?

(c) What is the domain of the function

h(x) = sin(\sqrt{1 − x}) ?

3. Evaluate the following limits. (a)

lim_{x→∞} \frac{sin(x) + x + 1}{2x + 1}

(b)

lim_{x→0} \frac{1 − (1 + x)^{3}}{x}

(c)

lim_{x→0} \frac{2 + 1/x}{\sqrt{1 + 4/x^{2}}}

4. The function

f(x) = \frac{x + 1}{x − 1}

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 δ > 0 such that for all x with 0 < |x − 1| < δ we have

|(x + 1)(x − 2) + 2| < \frac{1}{100}.

(b) Find an integer n such that f(x) = x^{3} − 5x^{2} − 4x − 5 has a zero between n and n + 1. As always, explain why your answer is true.