Q: Which of the following best explains why the function $$f(x) = \dfrac{x^2$ - 4}{x - 2}$$ has a removable discontinuity at $$x = 2$$?

The function is undefined at $$x = 2$$, but $$\lim_{x \to 2} f(x)$$ exists and equals 4.

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

$$\lim_{x \to a} f(x)$$ exists and equals $$f(a).$$

Q: Which of the following limits does NOT exist?

$$\lim_{x \to 0} \dfrac{1}{x}$$

Question 1

Evaluate the following expression: $$16^{-3/4}.$$

Accepted Answer

$$\dfrac{1}{8}$$

Question 2

Which of the following best explains why the function $$f(x) = \dfrac{x^2$ - 4}{x - 2}$$ has a removable discontinuity at $$x = 2$$?

Accepted Answer

The function is undefined at $$x = 2$$, but $$\lim_{x 	o 2} f(x)$$ exists and equals 4.

Question 3

Suppose the temperature $$T(t) (in $$degrees Celsius) in a laboratory is a continuous function of time $$t (in $$hours). If $$T(2) = 18$$ and $$T(5) = 24$$, what does the Intermediate Value Theorem guarantee?

Accepted Answer

There is a time $$c in$$ $$(2,5)$$ such that $$T(c) = 21.$$

Question 4

Given the graph of a function $$f(x)$$, which of the following statements is true if $$f(x)$$ has a sharp corner at $$x = 1$$?

Accepted Answer

$$f(x) is $$continuous but not differentiable at $$x = 1.$$

Question 5

Which of the following functions has a vertical asymptote at $$x = 3$$?

Accepted Answer

$$f(x) = \dfrac{1}{x-3}$$

Question 6

Evaluate the limit: $$\lim_{x 	o \infty} \dfrac{5x^2$ + 3}{2x^2$ - x}.$$

Accepted Answer

$$\dfrac{5}{2}$$

Question 7

Which of the following statements about asymptotes is TRUE?

Accepted Answer

A function can cross a horizontal asymptote.

Question 8

Let $$f(x) = \ln(x). $$What is the domain of $$f(x)$$?

Accepted Answer

$$(0, \infty)$$

Question 9

Given $$f(x) = \dfrac{2x+1}{x-4}$$, find all vertical and horizontal asymptotes.

Accepted Answer

Vertical: $$x=4$$; Horizontal: $$y=2$$

Question 10

Compute the derivative of $$f(x) = x^2$ at x = 3$ using the limit definition.

Accepted Answer

$$6$$

Question 11

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

Accepted Answer

$$\lim_{x 	o a} f(x)$$ exists and equals $$f(a).$$

Question 12

A car's position at time $$t is $$given by $$s(t) = 4t^2$ + 2t. $$What is the instantaneous velocity at $$t = 1$$?

Accepted Answer

$$10$$

Question 13

Which of the following best describes a jump discontinuity?

Accepted Answer

The left and right limits at a point exist but are not equal.

Question 14

Suppose $$f(x) is $$continuous on $$[1,5]$$ and $$f(1) = -2$$, $$f(5) = 6. $$Which value is guaranteed by the Intermediate Value Theorem to be achieved by $$f(x)$$ for some $$x in$$ $$[1,5]$$?

Accepted Answer

$$0$$

Question 15

Given $$f(x) = |x|$$, which statement is true about its differentiability?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 0.$$

Question 16

Find the equation of the tangent line to $$f(x) = x^2$ at x = 2$.

Accepted Answer

$$y = 4x - 4$$

Question 17

Which of the following is the correct interpretation of the derivative $$f'(a)$$?

Accepted Answer

The slope of the tangent line to $$f(x) at$$ $$x = a.$$

Question 18

Evaluate $$\log_2$ 8.$$

Accepted Answer

$$3$$

Question 19

Which of the following limits does NOT exist?

Accepted Answer

$$\lim_{x 	o 0} \dfrac{1}{x}$$

Calculus I Study Guide: Limits, Continuity, and the Derivative

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