Q: If $$\lim_{x \to a^-} f(x) = 5$$ and $$\lim_{x \to a^+} f(x) = 7$$, which of the following is true?

$$\lim_{x \to a} f(x)$$ does not exist

Q: If $$\lim_{x \to a^-} f(x) = +\infty$$ and $$\lim_{x \to a^+} f(x) = -\infty$$, what can be said about the two-sided limit $$\lim_{x \to a} f(x)$$?

$$\lim_{x \to a} f(x)$$ does not exist

Question 1

Which of the following best describes the meaning of $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ approaches the value $$L.$$

Question 2

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

Accepted Answer

$$4$$

Question 3

Evaluate $$\lim_{x 	o 2} \frac{|x-2|}{x-2}.$$

Accepted Answer

Does not exist

Question 4

Which of the following is the correct definition of a one-sided limit from the right?

Accepted Answer

$$\lim_{x 	o a^+} f(x) = L$$

Question 5

If $$f(x) = |x-1|$$, what is $$f(3)$$?

Accepted Answer

$$2$$

Question 6

Suppose $$f(x) is $$defined as $$f(x) = |x|. $$Which of the following is true for $$x \u003c 0$$?

Accepted Answer

$$f(x) = -x$$

Question 7

Evaluate $$\lim_{x 	o 3^-} \frac{|x^2$ - 9|}{x - 3}.$$

Accepted Answer

$$-6$$

Question 8

Evaluate $$\lim_{x 	o 3^+} \frac{|x^2$ - 9|}{x - 3}.$$

Accepted Answer

$$6$$

Question 9

Which of the following statements is true about the function $$f(x) = \frac{x^2$ - 4}{x - 2} at$$ $$x = 2$$?

Accepted Answer

The function has a removable discontinuity at $$x = 2.$$

Question 10

If $$\lim_{x 	o a^-} f(x) = 5$$ and $$\lim_{x 	o a^+} f(x) = 7$$, which of the following is true?

Accepted Answer

$$\lim_{x 	o a} f(x)$$ does not exist

Question 11

Which of the following best describes an infinite limit from the left?

Accepted Answer

As $$x$$ approaches $$a$$ from the left, $$f(x)$$ increases or decreases without bound.

Question 12

Given $$f(x) = \frac{1}{x}$$, what is $$\lim_{x 	o 0^-} f(x)$$?

Accepted Answer

$$-\infty$$

Question 13

Suppose $$f(x) is $$defined as $$f(x) = |x-1|. $$What is $$f(-2)$$?

Accepted Answer

$$3$$

Question 14

If $$\lim_{x 	o a^-} f(x) = +\infty$$ and $$\lim_{x 	o a^+} f(x) = -\infty$$, what can be said about the two-sided limit $$\lim_{x 	o a} f(x)$$?

Accepted Answer

$$\lim_{x 	o a} f(x)$$ does not exist

Question 15

A patient’s blood concentration of a drug is modeled by $$C(t) = \frac{10}{t-2}$$, where $$t is $$time in hours. What happens to $$C(t) as$$ $$t$$ approaches $$2$$ from the right?

Accepted Answer

$$C(t)$$ approaches $$+\infty$$

Question 16

Which of the following is the correct expression for the absolute value $$|x-1|$$?

Accepted Answer

$$x-1$$ for $$x \u003e 1$$, $$-(x-1)$$ for $$x \u003c 1$$

Question 17

If $$\lim_{x 	o a^-} f(x) = L_1$ and $$\lim_{x 	o a^+} f(x) = $$L_2$$ and $$L_1$$ 
eq $$L_2$$, what is true about $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f(x)$$ has a jump discontinuity at $$x = a$$

Question 18

Evaluate $$\lim_{x 	o 0^+} \frac{1}{x}.$$

Accepted Answer

$$+\infty$$

Question 19

A chemical reaction rate is modeled by $$R(c) = \frac{5}{c-1}$$, where $$c is $$the concentration of a reactant. What is $$\lim_{c 	o 1^-} R(c)$$?

Accepted Answer

$$-\infty$$

Limits and One-Sided Limits in Calculus: Definitions, Properties, and Examples

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