Q: Which theorem is illustrated by the second image, and what does it guarantee about the function $$g(x) as$$ $$x \to a$$?

The Squeeze Theorem; it guarantees $$\lim_{x \to a} g(x) = L if$$ $$f(x) \leq g(x) \leq h(x)$$ and $$\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L.$$

Q: The fourth image shows a function $$f(x)$$ with a vertical asymptote at $$x = a. $$Which of the following statements is true?

$$\lim_{x \to a} f(x) = \infty$$

Q: Based on the eighth image, is $$f(x)$$ continuous at $$x = 2$$? Justify your answer.

Yes, because $$\lim_{x \to 2} f(x) = f(2)$$ and both are defined and equal.

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

If $$-x^2$$ \leq $$x^2$$ \sin\left(\frac{1}{x}\right) \leq $$x^2$$ and $$\lim_{x \to 0} -x^2$ = \lim_{x \to 0} x^2$ = 0$$, then $$\lim_{x \to 0} x^2$ \sin\left(\frac{1}{x}\right) = 0$$

Question 1

Given the position function $$s(t)$$ shown in the first image, estimate the average velocity of the object over the interval $$[0.5, 2.5]$$ using the values from the graph.

Accepted Answer

$$\frac{150-46}{2.5-0.5} = 52$$ units/time

Question 2

Which theorem is illustrated by the second image, and what does it guarantee about the function $$g(x) as$$ $$x 	o a$$?

Accepted Answer

The Squeeze Theorem; it guarantees $$\lim_{x 	o a} g(x) = L if$$ $$f(x) \leq g(x) \leq h(x)$$ and $$\lim_{x 	o a} f(x) = \lim_{x 	o a} h(x) = L.$$

Question 3

In the third image, what is the value of $$\lim_{x 	o 0} x^2$ \sin\left(\frac{1}{x}ight)$$?

Accepted Answer

$$0$$

Question 4

The fourth image shows a function $$f(x)$$ with a vertical asymptote at $$x = a. $$Which of the following statements is true?

Accepted Answer

$$\lim_{x 	o a} f(x) = \infty$$

Question 5

In the fifth image, as $$x$$ approaches $$a$$, what happens to $$f(x)$$?

Accepted Answer

$$f(x) is $$negative and grows arbitrarily large in magnitude, so $$\lim_{x 	o a} f(x) = -\infty$$

Question 6

Refer to the sixth image. What are the equations of the vertical asymptotes for $$y = g(x)$$?

Accepted Answer

$$x = 2$$ and $$x = 4$$

Question 7

In the seventh image, what is the horizontal asymptote as $$x 	o \infty$$ for $$f(x)$$?

Accepted Answer

$$y = L$$

Question 8

Based on the eighth image, is $$f(x)$$ continuous at $$x = 2$$? Justify your answer.

Accepted Answer

Yes, because $$\lim_{x 	o 2} f(x) = f(2)$$ and both are defined and equal.

Question 9

Which type of discontinuity is present at $$x = 1 in $$the eighth image?

Accepted Answer

Infinite discontinuity

Question 10

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b). $$What theorem guarantees that $$f(x) = 0$$ has a solution in $$(a, b)$$?

Accepted Answer

Intermediate Value Theorem

Question 11

If $$\lim_{x 	o 2^-} f(x) = 3$$ and $$\lim_{x 	o 2^+} f(x) = 5$$, what can you conclude about $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

The limit does not exist

Question 12

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

Accepted Answer

If $$-x^2$$ \leq $$x^2$$ \sin\left(\frac{1}{x}ight) \leq $$x^2$$ and $$\lim_{x 	o 0} -x^2$ = \lim_{x 	o 0} x^2$ = 0$$, then $$\lim_{x 	o 0} x^2$ \sin\left(\frac{1}{x}ight) = 0$$

Question 13

If $$\lim_{x 	o a} f(x) = \infty$$, what does this mean about the function $$f(x)$$ near $$x = a$$?

Accepted Answer

$$f(x)$$ increases without bound as $$x$$ approaches $$a$$

Question 14

Which of the following best describes a removable discontinuity?

Accepted Answer

A hole in the graph where the limit exists but $$f(a) is $$either undefined or not equal to the limit

Question 15

If $$f(x) = \frac{1}{x^2$}$$, what is $$\lim_{x 	o \infty} f(x)$$ and what does this tell you about the horizontal asymptote?

Accepted Answer

$$0$$; the horizontal asymptote is $$y = 0$$

Limits, Continuity, and Exponential Functions: Study Notes for Algebra & Trigonometry

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