Q: Which of the following is the correct definition of the derivative of a function $$f at a $$point $$x_0$$?

$$f'(x_0$) = \lim\limits_{h \to 0} \frac{f(x_0$ + h) - f(x_0$)}{h}$$

Q: Suppose $$f(x) is $$continuous at $$x = 2. $$Which of the following must be true?

$$\lim\limits_{x \to 2} f(x)$$ exists and equals $$f(2)$$

Q: A rational function $$R(x) = \frac{P(x)}{Q(x)}$$ has a vertical asymptote at $$x = a if$$:

$$Q(a) = 0$$ and $$P(a) \neq 0$$

Q: Which of the following is the correct application of the Sandwich (Squeeze) Theorem?

If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim\limits_{x \to a} g(x) = \lim\limits_{x \to a} h(x) = L$$, then $$\lim\limits_{x \to a} f(x) = L$$

Question 1

Given the function $$f(x) = x^2$ - 4x$$, what is the slope of the tangent line to the curve at the point $$P(1, -3)$$ using the limit definition of the derivative?

Accepted Answer

$$-4$$

Question 2

Which of the following is the correct definition of the derivative of a function $$f at a $$point $$x_0$$?

Accepted Answer

$$f'(x_0$) = \lim\limits_{h 	o 0} \frac{f(x_0$ + h) - f(x_0$)}{h}$$

Question 3

Evaluate $$\lim\limits_{x 	o 1} \frac{x^2$ - 1}{x - 1}.$$

Accepted Answer

$$2$$

Question 4

Which of the following situations describes when a limit does NOT exist?

Accepted Answer

The function approaches different values from the left and right.

Question 5

Suppose $$f(x) is $$continuous at $$x = 2. $$Which of the following must be true?

Accepted Answer

$$\lim\limits_{x 	o 2} f(x)$$ exists and equals $$f(2)$$

Question 6

Given $$f(x) = \sqrt{x - 2}$$, for which values of $$x is$$ $$f(x)$$ continuous?

Accepted Answer

$$x \geq 2$$

Question 7

A rational function $$R(x) = \frac{P(x)}{Q(x)}$$ has a vertical asymptote at $$x = a if$$:

Accepted Answer

$$Q(a) = 0$$ and $$P(a) 
eq 0$$

Question 8

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 9

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

Accepted Answer

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

Question 10

A drug's effectiveness $$E(t)$$ increases for $$0  2. $$What can you say about $$E'(2)$$?

Accepted Answer

$$E'(2) = 0$$

Question 11

Which of the following is an example of a removable discontinuity?

Accepted Answer

A hole in the graph where the limit exists but the function is not defined

Question 12

Evaluate $$\lim\limits_{x 	o \infty} \frac{3x^2$ + 5x + 6}{x^2$ - 4}.$$

Accepted Answer

$$3$$

Question 13

If $$f(x) is $$differentiable at $$x = c$$, which of the following must also be true?

Accepted Answer

$$f(x) is $$continuous at $$x = c$$

Question 14

Which of the following is the correct application of the Sandwich (Squeeze) Theorem?

Accepted Answer

If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim\limits_{x 	o a} g(x) = \lim\limits_{x 	o a} h(x) = L$$, then $$\lim\limits_{x 	o a} f(x) = L$$

Question 15

Evaluate $$\lim\limits_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 16

Which of the following is a correct interpretation of the limit $$\lim\limits_{h 	o 0} \frac{f(x_0$ + h) - f(x_0$)}{h}$$?

Accepted Answer

The slope of the tangent line to $$y = f(x) at$$ $$x = x_0$

Question 17

Given $$f(x) = \frac{1}{x}$$, what is the equation of the tangent line at $$x = -1$$?

Accepted Answer

$$y = -x - 2$$

Question 18

Which of the following is NOT a situation where a function fails to be differentiable at a point?

Accepted Answer

The function is continuous and smooth at the point

Question 19

Evaluate $$\lim\limits_{x 	o 0} \frac{\sqrt{x + 1} - 1}{x}.$$

Accepted Answer

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

Calculus I: Limits, Continuity, and Derivatives – Study Notes

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