Q: Which of the following is the correct limit definition of the derivative of a function $$f(x) at$$ $$x = a$$?

$$\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Q: If $$f(x) = \begin{cases} x^2$, & x \u003c 1 \\ 2x - 1, & x \geq 1 \end{cases}$$, is $$f(x)$$ continuous at $$x = 1$$?

Yes, because $$\displaystyle \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)$$

Q: Which of the following limits does NOT exist?

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

Question 1

Which of the following is the correct limit definition of the derivative of a function $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\displaystyle \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 2

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ + 3, & x \u003c 4 \ 2x + k, & x \geq 4 \end{cases}. $$For what value of $$k is$$ $$f(x)$$ continuous everywhere?

Accepted Answer

$$k = 11$$

Question 3

Which of the following statements is NOT a requirement for a function $$f(x) to be $$continuous at $$x = a$$?

Accepted Answer

$$f(x) is $$differentiable at $$x = a$$

Question 4

Given $$f(x) = \sqrt{x}$$, what is the average rate of change of $$f(x) on $$the interval $$[1, 9]$$?

Accepted Answer

$$\dfrac{1}{2}$$

Question 5

If $$\displaystyle \lim_{x 	o 2^-} f(x) = 3$$ and $$\displaystyle \lim_{x 	o 2^+} f(x) = 5$$, which of the following is true?

Accepted Answer

$$f(x)$$ has a jump discontinuity at $$x = 2$$

Question 6

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

Accepted Answer

$$2x + 1$$

Question 7

Given $$f(x) = \dfrac{2x-4}{x-2}$$, what is $$\displaystyle \lim_{x 	o 2} f(x)$$?

Accepted Answer

$$2$$

Question 8

Which of the following functions has an infinite discontinuity at $$x = 0$$?

Accepted Answer

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

Question 9

Given $$f(x) = 3x^2$ - 4x + 5$$, what is the instantaneous rate of change at $$x = 1$$?

Accepted Answer

$$2$$

Question 10

Which of the following best describes the average rate of change of a function $$f(x)$$ over $$[a, b]$$?

Accepted Answer

$$\dfrac{f(b) - f(a)}{b - a}$$

Question 11

If $$f(x) = \begin{cases} x^2$, & x \u003c 1 \ 2x - 1, & x \geq 1 \end{cases}$$, is $$f(x)$$ continuous at $$x = 1$$?

Accepted Answer

Yes, because $$\displaystyle \lim_{x 	o 1^-} f(x) = \lim_{x 	o 1^+} f(x) = f(1)$$

Question 12

Evaluate $$\displaystyle \lim_{x 	o 0} \dfrac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 13

Given $$f(x) = \sqrt{x+1}$$, what is the instantaneous rate of change at $$x = 3$$?

Accepted Answer

$$\dfrac{1}{4}$$

Question 14

Which of the following is a removable discontinuity?

Accepted Answer

$$f(x) = \dfrac{x^2$ - 1}{x - 1} at$$ $$x = 1$$

Question 15

If $$f(x) = x^3$ - 2x + 1$$, what is $$f'(x)$$?

Accepted Answer

$$3x^2 - 2$

Question 16

Evaluate $$\displaystyle \lim_{x 	o 2} \dfrac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 17

Given $$f(x) = \dfrac{5x - 4}{x}$$, what is the derivative $$f'(x)$$?

Accepted Answer

$$\dfrac{4}{x^2$}$$

Question 18

Which of the following limits does NOT exist?

Accepted Answer

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

Question 19

If $$f(x) = x^2$ and g(x) = 2x$, what is $$\displaystyle \lim_{x 	o 1} [f(x) + g(x)]$$?

Accepted Answer

$$3$$

Calculus I Study Guide: Limits, Continuity, and Derivatives

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