Question 1

Evaluate $$\lim_{x 	o 0} \frac{1}{x^2$}. $$Which of the following best describes the behavior of the limit?

Accepted Answer

The limit is $$+\infty.$$

Question 2

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

Accepted Answer

$$0$$

Question 3

Evaluate $$\lim_{x 	o \infty} \frac{1}{x}.$$

Accepted Answer

$$0$$

Question 4

What is $$\lim_{x 	o 0^+} \frac{1}{x}$$?

Accepted Answer

$$+\infty$$

Question 5

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

Accepted Answer

$$+\infty$$

Question 6

Find $$\lim_{x 	o 0} \frac{\sin(3x)}{4x}.$$

Accepted Answer

$$\frac{3}{4}$$

Question 7

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

Accepted Answer

Does not exist (infinite)

Question 8

Calculate $$\lim_{x 	o 0} 7x^2$ + 2x + 1.$$

Accepted Answer

$$1$$

Question 9

Evaluate $$\lim_{t 	o 5} \frac{5 - \sqrt{t}}{t - 25}.$$

Accepted Answer

$$-\frac{1}{10}$$

Question 10

Suppose $$f(x) is $$defined everywhere except maybe $$x = -2$$, and $$\lim_{x 	o -2} f(x) = \infty. $$Which of the following best describes the behavior of $$f(x)$$ near $$x = -2$$?

Accepted Answer

The function increases without bound as $$x$$ approaches $$-2.$$

Question 11

Given $$f(-3) = 2$$ and $$f(1) = 3$$, which of the following statements is true?

Accepted Answer

$$f(-3) = 2$$ and $$f(1) = 3.$$

Question 12

If $$f'(3) = -4$$, what does this tell you about the graph of $$f(x) at$$ $$x = 3$$?

Accepted Answer

The tangent line at $$x = 3$$ has a slope of $$-4.$$

Question 13

If $$\lim_{x 	o 5^-} f(x) = -4$$ and $$\lim_{x 	o 5^+} f(x) = 5$$, what can you conclude about $$f(x) at$$ $$x = 5$$?

Accepted Answer

$$f(x)$$ has a jump discontinuity at $$x = 5.$$

Question 14

Which of the following is true about the function $$f(x) if$$ $$f is $$defined everywhere except maybe $$x = -2$$?

Accepted Answer

$$f(x) is $$defined for all $$x$$ except possibly $$x = -2.$$

Question 15

A chemical reaction rate $$R(t) is $$modeled by $$R(t) = 7t^2$ + 2t + 1. $$What is the instantaneous rate of change at $$t = 0$$?

Accepted Answer

$$2$$

Question 16

If $$\lim_{x 	o 0} \frac{\sin(x)}{x} = 1$$, what is $$\lim_{x 	o 0} \frac{\sin(3x)}{3x}$$?

Accepted Answer

$$1$$

Question 17

A patient’s medication concentration $$C(t) is $$given by $$C(t) = \frac{1}{t^2$}. $$What happens to the concentration as $$t$$ approaches $$0$$ from the right?

Accepted Answer

It approaches $$+\infty.$$

Question 18

If $$\lim_{x 	o 0} \frac{\cos(x)}{\sin(x)} is $$infinite, which of the following best explains why?

Accepted Answer

Because $$\sin(x)$$ approaches $$0$$ and $$\cos(x)$$ approaches $$1.$$

Question 19

Suppose $$f(x) is a $$function such that $$\lim_{x 	o a} f(x) = L$$ and $$f(a) is $$not defined. Which of the following is true?

Accepted Answer

The limit exists even though the function is not defined at $$x = a.$$

Calculus Study Notes: Limits and Graphical Analysis

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