Q: Which of the following is the correct application of the limit law for the sum of two functions $$f(x)$$ and $$g(x) as$$ $$x$$ approaches $$a$$?

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

Q: Which of the following statements about one-sided limits is correct?

$$\lim\limits_{x \to a} f(x)$$ exists if and only if $$\lim\limits_{x \to a^-} f(x) = \lim\limits_{x \to a^+} f(x)$$

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

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

Q: Which of the following is the correct application of the power law for limits?

$$\lim\limits_{x \to a} [f(x)]^n = [\lim\limits_{x \to a} f(x)]^n$$

Question 1

Which of the following is the correct application of the limit law for the sum of two functions $$f(x)$$ and $$g(x) as$$ $$x$$ approaches $$a$$?

Accepted Answer

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

Question 2

Evaluate $$\lim\limits_{x 	o 2} (3x^2$ - 4x + 1). $$Which of the following is correct?

Accepted Answer

$$3(2)^2 - 4(2) + 1 = 12 - 8 + 1 = 5$

Question 3

Which theorem allows you to compute $$\lim\limits_{x 	o a} [f(x)g(x)] as $$the product of the individual limits, provided both limits exist?

Accepted Answer

Limit Laws (Product Law)

Question 4

Given $$f(x) = \frac{x^2$ - 4}{x - 2}$$, what is $$\lim\limits_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

$$1$$

Question 7

Which technique is most appropriate for evaluating $$\lim\limits_{x 	o 3} \frac{x^2$ - 9}{x - 3}$$?

Accepted Answer

Factoring and cancelling common factors

Question 8

Evaluate $$\lim\limits_{x 	o 1} \frac{\sqrt{x} - 1}{x - 1}$$ using the conjugate method.

Accepted Answer

$$\frac{1}{2}$$

Question 9

If $$\lim\limits_{x 	o a} f(x) = L$$ and $$\lim\limits_{x 	o a} g(x) = M$$, what is $$\lim\limits_{x 	o a} [f(x) - g(x)]$$?

Accepted Answer

$$L - M$$

Question 10

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

Accepted Answer

$$\lim\limits_{x 	o a} f(x)$$ exists if and only if $$\lim\limits_{x 	o a^-} f(x) = \lim\limits_{x 	o a^+} f(x)$$

Question 11

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

Accepted Answer

$$0$$

Question 12

Which of the following is NOT a valid limit law?

Accepted Answer

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

Question 13

A chemist is studying the reaction rate $$R(t) = 5t^2$ - 3t + 2. $$What is the instantaneous rate as $$t$$ approaches $$1$$?

Accepted Answer

$$5(1)^2 - 3(1) + 2 = 5 - 3 + 2 = 4$

Question 14

Which of the following best describes the use of factoring to compute limits?

Accepted Answer

Factoring is used to cancel common factors in the numerator and denominator to resolve indeterminate forms.

Question 15

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin(3x)}{x}.$$

Accepted Answer

$$3$$

Question 16

A medical researcher models the concentration of a drug in the bloodstream as $$C(t) = \frac{2t^2$ + 3t + 1}{t + 1}. $$What is $$\lim\limits_{t 	o 1} C(t)$$?

Accepted Answer

$$\frac{2(1)^2$ + 3(1) + 1}{1 + 1} = \frac{2 + 3 + 1}{2} = \frac{6}{2} = 3$$

Question 17

Which of the following is the correct application of the power law for limits?

Accepted Answer

$$\lim\limits_{x 	o a} [f(x)]^n = [\lim\limits_{x 	o a} f(x)]^n$$

Question 18

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

Accepted Answer

$$2$$

Techniques for Computing Limits in Calculus

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