Q: Which of the following is a correct application of the Squeezing Theorem?

If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$$, then $$\lim_{x \to c} f(x) = L$$

Question 1

A skydiver jumps from a plane and eventually reaches a constant speed called terminal velocity. Which calculus concept best describes why the skydiver's velocity stops increasing?

Accepted Answer

The concept of a limit

Question 2

Which of the following best describes the limit $$\lim_{x 	o 2} (x^2$ - x + 1)$$?

Accepted Answer

$$3$$

Question 3

The tangent line to the curve $$y = x^2$ at the point (1$$, 1)$$ has which equation?

Accepted Answer

$$y = 2x - 1$$

Question 4

Which of the following is true about the limit $$\lim_{x 	o 0} \frac{\sin x}{x}$$?

Accepted Answer

The limit is $$1$$ because $$\sin x$$ and $$x$$ approach $$0 at $$the same rate.

Question 5

A function $$f(x) is $$defined as $$f(x) = |x|/x. $$What is $$\lim_{x 	o 0^-} f(x)$$?

Accepted Answer

$$-1$$

Question 6

Which of the following statements about two-sided limits is correct?

Accepted Answer

A two-sided limit exists if and only if both one-sided limits exist and are equal.

Question 7

Suppose $$\lim_{x 	o 0^+} \frac{1}{x} = +\infty. $$What does this mean about the graph of $$y = \frac{1}{x}$$?

Accepted Answer

The graph increases without bound as $$x$$ approaches $$0$$ from the right

Question 8

Which of the following is a necessary condition for a function $$f(x) to be $$continuous at $$x = c$$?

Accepted Answer

All of the above

Question 9

A cable is accidentally cut at time $$t_0$$, causing the voltage to drop instantly to zero. What type of discontinuity does the voltage function have at $$t_0$$?

Accepted Answer

Jump discontinuity

Question 10

Which of the following best describes the end behavior of the function $$f(x) = \frac{3x + 5}{6x - 8} as$$ $$x 	o +\infty$$?

Accepted Answer

$$f(x) 	o 1/2$$

Question 11

Which of the following is the correct formal definition of $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that $$|f(x) - L| \u003c \epsilon$$ whenever $$0 \u003c |x - a| \u003c \delta$$

Question 12

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

Accepted Answer

A function with a hole that can be filled to make it continuous

Question 13

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b)$$, what does the Intermediate Value Theorem guarantee?

Accepted Answer

There is at least one $$c in$$ $$(a, b)$$ such that $$f(c) = 0$$

Question 14

Which of the following is true about the function $$f(x) = \sin x as$$ $$x 	o +\infty$$?

Accepted Answer

$$f(x)$$ oscillates between $$-1$$ and $$1$$ and does not approach a limit

Question 15

Which of the following is a correct application of the Squeezing Theorem?

Accepted Answer

If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x 	o c} g(x) = \lim_{x 	o c} h(x) = L$$, then $$\lim_{x 	o c} f(x) = L$$

Question 16

Which of the following is true about the continuity of the function $$f(x) = \frac{x^2$ - 9}{x^2$ - 5x + 6}$$?

Accepted Answer

It is discontinuous at $$x = 2$$ and $$x = 3$$

Question 17

Which of the following is the correct statement about the continuity of the absolute value function $$|x|$$?

Accepted Answer

$$|x| is $$continuous everywhere

Question 18

Suppose $$f(x) is $$continuous everywhere and $$g(x) is $$continuous everywhere. What can you say about $$|g(x)|$$?

Accepted Answer

$$|g(x)| is $$continuous everywhere

Question 19

Which of the following is a correct statement about the limit $$\lim_{x 	o 0} x \sin(1/x)$$?

Accepted Answer

The limit is $$0 by $$the Squeezing Theorem

Limits and Continuity: Foundations of Calculus

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