Question 1

Evaluate the following limit: $$\lim_{x 	o 2} \frac{x^2$ + x - 6}{x^2$ + 4x - 12}$$

Accepted Answer

$$\frac{5}{4}$$

Question 2

For which value of $$a$$ does the limit $$\lim_{x 	o 3} \frac{5x - 9}{x - a}$$ exist?

Accepted Answer

$$a 
eq 3$$

Question 3

Suppose $$f(x) = \begin{cases} 2x^2$ + ax & x \u003c 1 \ bx + 4 & x \geq 1 \end{cases}. If$$ $$f is $$continuous at $$x = 1$$, what are the values of $$a$$ and $$b$$?

Accepted Answer

$$a = 2, b = 6$$

Question 4

Evaluate $$\lim_{x 	o 0} \frac{\sin(2x)}{x}.$$

Accepted Answer

$$2$$

Question 5

Determine $$\lim_{x 	o \infty} \frac{16 - x^2$}{x^2$ + 4}.$$

Accepted Answer

$$0$$

Question 6

Find $$\lim_{x 	o 1} \frac{|x - 1|}{x^2$ - 1}.$$

Accepted Answer

$$\frac{1}{2}$$ from the right, $$-\frac{1}{2}$$ from the left

Question 7

Evaluate $$\lim_{x 	o 0} \frac{	an(ax - a) + b}{x - 1}$$, given that the limit equals $$3. $$What is the value of $$b$$?

Accepted Answer

$$b = 2 + a$$

Question 8

A fish moves along the $$y-$$axis according to $$s(t) = \frac{1}{2}t^2$ - 2\sqrt{t}$$, where $$t is in $$minutes and $$s in $$meters. What is the instantaneous velocity at $$t = 4$$?

Accepted Answer

$$4 - \frac{2}{\sqrt{4}}$$

Question 9

Evaluate $$\lim_{x 	o 0} \frac{1 - \cos x}{x^2$}.$$

Accepted Answer

$$\frac{1}{2}$$

Question 10

Suppose $$\lim_{x 	o a} f(x) = A$$ and $$\lim_{x 	o a} f(x^2$) = B. $$Which of the following is always true?

Accepted Answer

$$B = f(a^2$) if$$ $$f is $$continuous$

Question 11

Evaluate $$\lim_{x 	o 0} \frac{\sin(4x)}{x}.$$

Accepted Answer

$$4$$

Question 12

Find the average velocity of the fish described by $$s(t) = \frac{1}{2}t^2$ - 2\sqrt{t}$$ between $$t = 0$$ and $$t = 4.$$

Accepted Answer

$$\frac{s(4) - s(0)}{4}$$

Question 13

Evaluate $$\lim_{x 	o 0} \frac{\sin^2$(x)}{x^2$}.$$

Accepted Answer

$$1$$

Question 14

Evaluate $$\lim_{x 	o \infty} \frac{5 - \sqrt{3x^2$ - 25}}{2x^2$ + 2x^2$ - 21x}.$$

Accepted Answer

$$0$$

Question 15

If $$\lim_{x 	o 0} \frac{\sqrt{x + 7} - \sqrt{7}}{x}$$ exists, what is its value?

Accepted Answer

$$\frac{1}{2\sqrt{7}}$$

Question 16

Suppose $$\lim_{x 	o a} f(x) = 2$$ and $$\lim_{x 	o a} \frac{f(x) - 2}{x - a} = 5. $$What is $$f'(a)$$?

Accepted Answer

$$5$$

Question 17

Evaluate $$\lim_{x 	o 0} \frac{\sin(x)}{x}.$$

Accepted Answer

$$1$$

Question 18

If $$f(x) is $$continuous at $$x = 1$$, which of the following must be true?

Accepted Answer

$$\lim_{x 	o 1} f(x) = f(1)$$

Question 19

Evaluate $$\lim_{x 	o 0} \frac{\sqrt{x + 3} - \sqrt{3}}{x}.$$

Accepted Answer

$$\frac{1}{2\sqrt{3}}$$

Calculus I: Limits and Continuity – Problem Set Study Guide

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