Math 161 §2, Calculus 1 , Fall 97
Exam 1

Name:

Answer all questions.

1.

Define: $${\lim_{x\to a} f(x) =L}$$ (10 Points)

2.

Prove that $${\lim_{x\to 3}\frac{3x^2-10x +3}{x-3}=8}$$

(a)

Using the $\delta$-$\epsilon$ definition (10 Points)

(b)

Using limit theorems (10 Points)

3.

Say which limit theorems you are using as you find

(a)

$${\lim_{x\to 2} \sqrt{\frac{x^2 +12}{x^3 +1}}}$$(10 Points)

(b)

$${\lim_{x\to 0} x^2 \sin(\frac1x)}$$(10 Points)

(c)

$${\lim_{x\to 0} g(x)}$$ if $${g(x) =\left\{\begin{array}{ll} x^2 & \mbox{ if } x<0 \\ 5 & \mbox{ if }x=0 \\ 2x & \mbox{ if } x>0 \end{array} \right. }$$(10 Points)

4.

Find the following limits (no justification necessary):

(a)

$${ \lim_{h\to 0} \frac{\frac1{2+h} - \frac12}{h}}$$(10 Points)

(b)

$${\lim_{h\to 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}}$$(10 Points)

5.

Sketch graphs of functions which fail to be continuous at 2 because (10 Points)

(a)

$${\lim_{x\to 2} f(x)}$$ does not exist

(b)

$${\lim_{x\to 2} f(x) \neq f(2)}$$

6.

Prove that if $${\lim_{x\to a} f(x) =L \mbox{ and } \lim_{x\to a}g(x) =M}$$ then $${\lim_{x\to a} (f+g)(x) = L+M}$$.
(10 Points)

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