Q: If $$\lim_{x \to 2^-} f(x) = 3$$ and $$\lim_{x \to 2^+} f(x) = 5$$, what can you conclude about $$\lim_{x \to 2} f(x)$$?

$$\lim_{x \to 2} f(x)$$ does not exist

Q: Which of the following is an example of a function where the limit does not exist due to a jump?

$$f(x) = \left\{ \begin{array}{ll} 1, & x \u003c 0 \\ 2, & x \geq 0 \end{array} \right.$$

Question 1

Suppose an object is thrown upward from the ground with a speed of $$v_0 = 96$ ft/s. The height after t$ seconds is given by s(t) = 96t$$ - $$16t^2. $$What is the instantaneous velocity at $$t = 2$$ seconds?

Accepted Answer

$$32 ft/s$$

Question 2

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

Accepted Answer

As $$x$$ gets arbitrarily close to $$a$$, $$f(x)$$ gets arbitrarily close to $$L.$$

Question 3

Given the function $$f(x) = \left\{ \begin{array}{ll} x, & x \geq 0 \ -x, & x \u003c 0 \end{array} ight.$$, what is $$\lim_{x 	o 0^-} f(x)$$?

Accepted Answer

$$-0$$

Question 4

Which of the following is NOT a reason why a limit might fail to exist?

Accepted Answer

The function is continuous at $$x = a.$$

Question 5

If $$\lim_{x 	o 2^-} f(x) = 3$$ and $$\lim_{x 	o 2^+} f(x) = 5$$, what can you conclude about $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$\lim_{x 	o 2} f(x)$$ does not exist

Question 6

A function $$f(x) is $$defined as $$f(x) = \frac{1}{x-2}. $$What happens to $$f(x) as$$ $$x$$ approaches $$2$$?

Accepted Answer

$$f(x)$$ increases or decreases without bound$

Question 7

Given the graph of a function, how can you estimate $$\lim_{x 	o a} f(x)$$?

Accepted Answer

By observing the $$y-$$values as $$x$$ approaches $$a$$ from both sides

Question 8

Suppose $$f(x) is $$defined for all $$x$$ except possibly at $$x = 1. If$$ $$f(x)$$ gets arbitrarily close to $$5 as$$ $$x$$ gets arbitrarily close to $$1$$, what is $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

$$5$$

Question 9

Which of the following statements is TRUE about the relationship between one-sided and two-sided limits?

Accepted Answer

The two-sided limit exists if and only if both one-sided limits exist and are equal.

Question 10

If $$f(x) is $$defined as $$f(x) = \sin\left(\frac{1}{x}\right)$$, what happens to $$f(x) as$$ $$x$$ approaches $$0$$?

Accepted Answer

$$f(x)$$ does not exist due to oscillation$

Question 11

A car's position is given by $$s(t) = t^2$ + 2t. $$What is the average velocity between $$t = 1$$ and $$t = 3$$?

Accepted Answer

$$6$$

Question 12

Which of the following is the correct definition of the left-sided limit $$\lim_{x 	o a^-} f(x)$$?

Accepted Answer

The value $$f(x)$$ approaches as $$x$$ approaches $$a$$ from values less than $$a.$$

Question 13

If $$f(x) is $$defined as $$f(x) = |x|$$, what is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$0$$

Question 14

Which of the following is an example of a function where the limit does not exist due to a jump?

Accepted Answer

$$f(x) = \left\{ \begin{array}{ll} 1, & x \u003c 0 \ 2, & x \geq 0 \end{array} ight.$$

Question 15

If $$\lim_{x 	o 3} f(x) = 7$$, which of the following must be true?

Accepted Answer

$$f(x)$$ gets arbitrarily close to $$7 as$$ $$x$$ gets close to $$3$$

Question 16

Suppose $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 1}{x - 1}$$ for $$x 
eq 1. $$What is $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

$$2$$

Introduction to Limits: Concepts, Definitions, and Applications

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