Q: Which of the following is the correct application of the Sum Rule for limits, given $$\lim\limits_{x \to c} f(x) = L$$ and $$\lim\limits_{x \to c} g(x) = M$$?

$$\lim\limits_{x \to c} [f(x) + g(x)] = L + M$$

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

$$\lim\limits_{x \to c} [f(x)g(x)] = \lim\limits_{x \to c} f(x) + \lim\limits_{x \to c} g(x)$$

Question 1

Which of the following is the correct application of the Sum Rule for limits, given $$\lim\limits_{x 	o c} f(x) = L$$ and $$\lim\limits_{x 	o c} g(x) = M$$?

Accepted Answer

$$\lim\limits_{x 	o c} [f(x) + g(x)] = L + M$$

Question 2

Evaluate $$\lim\limits_{x 	o 0} \dfrac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 3

Given $$\lim\limits_{x 	o 2} p(x) = 4$$ and $$\lim\limits_{x 	o 2} v(x) = 0$$, find $$\lim\limits_{x 	o 2} [-4p(x) + 5v(x)].$$

Accepted Answer

$$-16$$

Question 4

Which theorem allows you to conclude $$\lim\limits_{x 	o c} f(x) = L if$$ $$g(x) \leq f(x) \leq h(x)$$ and $$\lim\limits_{x 	o c} g(x) = \lim\limits_{x 	o c} h(x) = L$$?

Accepted Answer

The Sandwich Theorem

Question 5

Evaluate $$\lim\limits_{x 	o 0} \dfrac{1 - \cos x}{x}.$$

Accepted Answer

$$0$$

Question 6

If $$\lim\limits_{x 	o 2} f(x) = 5$$ and $$\lim\limits_{x 	o 2} g(x) = -3$$, what is $$\lim\limits_{x 	o 2} [f(x)g(x)]$$?

Accepted Answer

$$-15$$

Question 7

Evaluate $$\lim\limits_{x 	o 0} x^2$ \sin\left(\dfrac{1}{x}ight)$$ using the Sandwich Theorem.

Accepted Answer

$$0$$

Question 8

Which of the following is an indeterminate form?

Accepted Answer

$$\dfrac{0}{0}$$

Question 9

Given $$\lim\limits_{x 	o 2} f(x) = 5$$ and $$\lim\limits_{x 	o 2} g(x) = -3$$, find $$\lim\limits_{x 	o 2} \dfrac{f(x)}{g(x)}.$$

Accepted Answer

$$\dfrac{5}{-3}$$

Question 10

Evaluate $$\lim\limits_{x 	o 0} \dfrac{x}{|x|}.$$

Accepted Answer

The limit does not exist

Question 11

Suppose $$\lim\limits_{x 	o 2} f(x) = 5$$ and $$\lim\limits_{x 	o 2} g(x) = -3. $$What is $$\lim\limits_{x 	o 2} [2f(x) + 3g(x)]$$?

Accepted Answer

$$1$$

Question 12

Which of the following is NOT a valid limit law?

Accepted Answer

$$\lim\limits_{x 	o c} [f(x)g(x)] = \lim\limits_{x 	o c} f(x) + \lim\limits_{x 	o c} g(x)$$

Question 13

Evaluate $$\lim\limits_{x 	o 0} \dfrac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 14

If $$\lim\limits_{x 	o 2} f(x) = 5$$ and $$\lim\limits_{x 	o 2} g(x) = -3$$, what is $$\lim\limits_{x 	o 2} [f(x) - g(x)]$$?

Accepted Answer

$$8$$

Question 15

Which of the following best describes the behavior of $$f(x) = \sin\left(\dfrac{1}{x}ight) as$$ $$x 	o 0$$?

Accepted Answer

The function oscillates infinitely often as $$x$$ approaches $$0$$ and the limit does not exist.

Question 16

Evaluate $$\lim\limits_{x 	o 0} \dfrac{1 - \cos x}{x^2$}.$$

Accepted Answer

$$\dfrac{1}{2}$$

Limits and Limit Laws in Calculus: Study Notes and Examples

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