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

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

Q: Which of the following is NOT a valid limit law?

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$$ for all $$g(x)$$

Question 1

Which of the following is the correct application of the limit laws to evaluate $$\lim_{x 	o a} (7x^3$ + 3x^2$ + 4x + 2)$$?

Accepted Answer

$$7a^3$$ + $$3a^2 + 4a + 2$

Question 2

Given $$\lim_{x 	o 2} f(x) = 4$$, $$\lim_{x 	o 2} g(x) = 5$$, and $$\lim_{x 	o 2} h(x) = 8$$, what is $$\lim_{x 	o 2} \frac{f(x) - g(x)}{h(x)}$$?

Accepted Answer

$$\frac{4 - 5}{8} = -\frac{1}{8}$$

Question 3

Evaluate $$\lim_{x 	o 1} \frac{5x^4$ - 3x^2$ + 8x - 6}{x + 1}.$$

Accepted Answer

$$\frac{5(1)^4$ - 3(1)^2$ + 8(1) - 6}{1 + 1} = \frac{5 - 3 + 8 - 6}{2} = \frac{4}{2} = 2$$

Question 4

Which theorem allows you to conclude that $$\lim_{x 	o a} p(x) = p(a)$$ for any polynomial $$p(x)$$?

Accepted Answer

Theorem 2.2 (Limit Laws for Polynomials)

Question 5

Suppose $$f(x) = \begin{cases} 0 & x \leq -5 \ \sqrt{25 - x^2$} & -5 \u003c x \u003c 5 \ 3x & x \geq 5 \end{cases}. $$What is $$\lim_{x 	o 5^-} f(x)$$?

Accepted Answer

$$0$$

Question 6

For the same function $$f(x) as $$above, what is $$\lim_{x 	o 5^+} f(x)$$?

Accepted Answer

$$15$$

Question 7

For the same function $$f(x) as $$above, what is $$\lim_{x 	o 5} f(x)$$?

Accepted Answer

Does not exist

Question 8

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

Accepted Answer

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

Question 9

Use the Squeeze Theorem to evaluate $$\lim_{x 	o 0} x^2$ \sin\left(\frac{1}{x}ight).$$

Accepted Answer

$$0$$

Question 10

Evaluate $$\lim_{x 	o 0} \frac{\sin^2$ x}{1 - \cos x}.$$

Accepted Answer

$$2$$

Question 11

Evaluate $$\lim_{x 	o 0} \frac{1 - \cos 2x}{\sin x}.$$

Accepted Answer

Question 12

Which of the following is NOT a valid limit law?

Accepted Answer

$$\lim_{x 	o a} \frac{f(x)}{g(x)} = \frac{\lim_{x 	o a} f(x)}{\lim_{x 	o a} g(x)}$$ for all $$g(x)$$

Question 13

If $$\lim_{x 	o a} f(x) = 3$$ and $$\lim_{x 	o a} g(x) = -2$$, what is $$\lim_{x 	o a} (5f(x) - 2g(x))$$?

Accepted Answer

$$5 \cdot 3 - 2 \cdot (-2) = 15 + 4 = 19$$

Question 14

Which of the following is a necessary condition for applying the quotient law for limits?

Accepted Answer

$$\lim_{x 	o a} g(x) 
eq 0$$

Question 15

Evaluate $$\lim_{x 	o 2} (6f(x)g(x) + h(x))$$ given $$\lim_{x 	o 2} f(x) = 4$$, $$\lim_{x 	o 2} g(x) = 5$$, $$\lim_{x 	o 2} h(x) = 8.$$

Accepted Answer

$$6 \cdot 4 \cdot 5 + 8 = 120 + 8 = 128$$

Question 16

Which of the following is true about one-sided limits for even roots?

Accepted Answer

The function inside the root must be non-negative near $$a$$ from the appropriate side.

Calculus I: Techniques for Computing Limits

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