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: Using the eighth image, is $$f(x)$$ continuous at $$x = 2$$?

Yes, because $$\lim_{x \to 2} f(x) = f(2)$$

Q: Which of the following is a correct application of the Squeeze Theorem for $$\lim_{x \to 0} x^2$ \sin\left(\frac{1}{x}\right)$$?

Since $$-x^2$$ \leq $$x^2$$ \sin\left(\frac{1}{x}\right) \leq $$x^2$$ and $$\lim_{x \to 0} -x^2$ = 0 = \lim_{x \to 0} x^2$, the limit is 0.

Question 1

Given the position function $$s(t)$$ shown in the first image, estimate the average velocity of the object over the interval $$[1, 2.5]$$ using the values from the graph.

Accepted Answer

$$\frac{150 - 84}{2.5 - 1} = \frac{66}{1.5} = 44$$ units/time

Question 2

Which of the following best describes the Squeeze Theorem as illustrated in the second image?

Accepted Answer

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

Question 3

Refer to the third image. What is $$\lim_{x 	o 0} x^2$ \sin\left(\frac{1}{x}ight)$$?

Accepted Answer

0, by the Squeeze Theorem

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. Which $$x-$$values are vertical asymptotes for $$y = g(x)$$?

Accepted Answer

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

Question 7

The seventh image illustrates limits at infinity. What are the horizontal asymptotes of $$f(x)$$?

Accepted Answer

$$y = L$$ and $$y = M$$

Question 8

Using the eighth image, is $$f(x)$$ continuous at $$x = 2$$?

Accepted Answer

Yes, because $$\lim_{x 	o 2} f(x) = f(2)$$

Question 9

Based on the eighth image, which type of discontinuity is present at $$x = 1$$?

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

Accepted Answer

$$x = 2 is a $$vertical asymptote

Question 12

Which of the following is a correct application of the Squeeze Theorem for $$\lim_{x 	o 0} x^2$ \sin\left(\frac{1}{x}ight)$$?

Accepted Answer

Since $$-x^2$$ \leq $$x^2$$ \sin\left(\frac{1}{x}ight) \leq $$x^2$$ and $$\lim_{x 	o 0} -x^2$ = 0 = \lim_{x 	o 0} x^2$, the limit is 0.

Question 13

If $$\lim_{x 	o a} f(x) = L$$ and $$f(a) is $$not defined, what type of discontinuity does $$f$$ have at $$x = a$$?

Accepted Answer

Removable discontinuity

Limits and Continuity: Foundations for Calculus

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