Q: Suppose $$f(x)$$ has a jump discontinuity at $$x=1. $$Which of the following is TRUE?

$$\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x)$$

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

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

Question 1

Evaluate the following limit analytically: $$\lim_{x 	o 0} \frac{\sqrt{x+3} - \sqrt{3}}{x}. $$What is the value of the limit?

Accepted Answer

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

Question 2

Which of the following statements is TRUE about the function $$f(x) = \frac{1}{(x-2)^2$}$$?

Accepted Answer

It is continuous everywhere except at $$x=2$$ where it has a non-removable discontinuity.

Question 3

Given $$g(x) = \frac{x-2}{x^2$+2x-6}$$, at which $$x-$$value does $$g(x)$$ have a removable discontinuity?

Accepted Answer

$$x=2$$

Question 4

Find $$\lim_{x 	o 3} \frac{x+1-2}{x-3}.$$

Accepted Answer

Question 5

A medication's concentration in the bloodstream is modeled by $$C(t) = \frac{5t}{t^2$+1}. $$What is $$\lim_{t 	o \infty} C(t)$$?

Accepted Answer

$$0$$

Question 6

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$-11x-6 & x \geq -4 \ 2x & x \u003c -4 \end{cases}. $$What is $$\lim_{x 	o -4^-} f(x)$$?

Accepted Answer

$$-8$$

Question 7

Which of the following 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 8

Find the slope of the tangent line to $$f(x) = 3x^2$ + x - 4 at$$ $$x = -1$$ using the limit definition of derivative.

Accepted Answer

$$-5$$

Question 9

Given the graph of $$f(x)$$ below, which of the following $$x-$$values is a point of non-removable discontinuity?

Accepted Answer

$$x=2 ($$vertical asymptote)

Question 10

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

Accepted Answer

$$1$$

Question 11

A car's position is given by $$s(t) = -16t^2$ + 100t. $$What is the average velocity over the interval $$[3,5]$$?

Accepted Answer

$$-20$$

Question 12

Which of the following limits does NOT exist?

Accepted Answer

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

Question 13

For $$f(x) = \frac{x^2$-13x+30}{x^2$-100x}$$, which $$x-$$values are holes (removable discontinuities)?

Accepted Answer

$$x=10$$

Question 14

Find $$\lim_{x 	o 0} \frac{4(\sqrt{x+2})-2}{x}.$$

Accepted Answer

$$1$$

Question 15

Which of the following is the correct equation for the tangent line to $$y = x^2$ - 9 at $$the point $$(2, -5)$$?

Accepted Answer

$$y = 4x - 13$$

Question 16

Suppose $$f(x)$$ has a jump discontinuity at $$x=1. $$Which of the following is TRUE?

Accepted Answer

$$\lim_{x 	o 1^-} f(x) 
eq \lim_{x 	o 1^+} f(x)$$

Question 17

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

Accepted Answer

$$1$$

Question 18

A function $$f(x) is $$continuous everywhere except at $$x=3$$, where $$\lim_{x 	o 3} f(x)$$ exists but $$f(3) is $$not defined. What type of discontinuity is this?

Accepted Answer

Removable discontinuity

Question 19

Find $$\lim_{x 	o 0} \frac{\cos x - 1}{x}.$$

Accepted Answer

$$0$$

Question 20

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

Accepted Answer

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

Limits and Continuity: Study Guide for Calculus I

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