Q: Which statement about one-sided limits is TRUE?

$$\lim\limits_{x \to a^+} f(x) is $$the limit as $$x$$ approaches $$a$$ from the right.

Q: According to Theorem 2.1, when does $$\lim\limits_{x \to a} f(x)$$ exist?

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

Q: Suppose $$f(x) is $$not defined at $$x=a. $$Which of the following is TRUE?

$$\lim\limits_{x \to a} f(x)$$ can exist even if $$f(a) is $$undefined.

Q: Using the graph of $$y = f(x)$$, you observe that as $$x$$ approaches $$3$$ from the left, $$f(x)$$ approaches $$2$$, and as $$x$$ approaches $$3$$ from the right, $$f(x)$$ approaches $$4. $$What can you conclude?

$$\lim\limits_{x \to 3} f(x)$$ does not exist.

Q: Which of the following statements is FALSE?

The value of $$\lim\limits_{x \to a} f(x) is $$always found by computing $$f(a).$$

Q: If $$\lim\limits_{x \to a^-} f(x) = L_1$ and $$\lim\limits_{x \to a^+} f(x) = $$L_2$$ with $$L_1$$ \neq $$L_2$$, what can you say about $$\lim\limits_{x \to a} f(x)$$?

$$\lim\limits_{x \to a} f(x)$$ does not exist.

Question 1

Suppose $$f(x) is $$defined for all $$x$$ near $$a$$ except possibly at $$a. $$Which of the following best describes $$\lim\limits_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$ from both sides, $$f(x)$$ gets arbitrarily close to $$L$$, regardless of $$f(a).$$

Question 2

Given the function $$f(x) = \frac{x^2$ - 8}{4x - 2}$$, use the table of values near $$x=2 to $$estimate $$\lim\limits_{x 	o 2} f(x). $$Which of the following is closest to the correct value?

Accepted Answer

$$2.5$$

Question 3

Which statement about one-sided limits is TRUE?

Accepted Answer

$$\lim\limits_{x 	o a^+} f(x) is $$the limit as $$x$$ approaches $$a$$ from the right.

Question 4

According to Theorem 2.1, when does $$\lim\limits_{x 	o a} f(x)$$ exist?

Accepted Answer

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

Question 5

Given the graph of $$f(x)$$ below, $$f(2) is $$defined and equals $$3$$, but $$\lim\limits_{x 	o 2} f(x) = 5. $$What does this illustrate?

Accepted Answer

The value of $$f(2)$$ does not affect the value of $$\lim\limits_{x 	o 2} f(x).$$

Question 6

Evaluate $$\lim\limits_{x 	o 1} \frac{\sqrt{x} - 1}{x - 1}$$ using a table of values. What is the correct value?

Accepted Answer

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

Question 7

A function $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ + 1 & 	ext{if } x \leq -1 \ 3 & 	ext{if } x \u003e -1 \end{cases}. $$What is $$\lim\limits_{x 	o -1} f(x)$$?

Accepted Answer

The limit does not exist.

Question 8

Suppose $$f(x) is $$not defined at $$x=a. $$Which of the following is TRUE?

Accepted Answer

$$\lim\limits_{x 	o a} f(x)$$ can exist even if $$f(a) is $$undefined.

Question 9

Using the graph of $$y = f(x)$$, you observe that as $$x$$ approaches $$3$$ from the left, $$f(x)$$ approaches $$2$$, and as $$x$$ approaches $$3$$ from the right, $$f(x)$$ approaches $$4. $$What can you conclude?

Accepted Answer

$$\lim\limits_{x 	o 3} f(x)$$ does not exist.

Question 10

Evaluate $$\lim\limits_{h 	o 0} \frac{\sin h}{h}$$ using your knowledge of limits and the unit circle.

Accepted Answer

$$1$$

Question 11

Which of the following statements is FALSE?

Accepted Answer

The value of $$\lim\limits_{x 	o a} f(x) is $$always found by computing $$f(a).$$

Question 12

Given $$f(x) = \frac{x^2$ + x - 2}{x - 1}$$, what is $$\lim\limits_{x 	o 1} f(x)$$?

Accepted Answer

$$3$$

Question 13

Suppose $$f(x) is $$defined for all $$x$$ near $$a$$ except at $$a$$, and $$\lim\limits_{x 	o a} f(x)$$ exists. Which of the following must be true?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ approaches a single value from both sides.

Question 14

If $$\lim\limits_{x 	o a^-} f(x) = L_1$ and $$\lim\limits_{x 	o a^+} f(x) = $$L_2$$ with $$L_1$$ 
eq $$L_2$$, what can you say about $$\lim\limits_{x 	o a} f(x)$$?

Accepted Answer

$$\lim\limits_{x 	o a} f(x)$$ does not exist.

Question 15

Which of the following is a real-world example where a limit is used to analyze a physical phenomenon?

Accepted Answer

Calculating the instantaneous speed of a car by finding the limit of average speed as the time interval approaches zero.

Question 16

Evaluate $$\lim\limits_{x 	o 0} \cot x.$$

Accepted Answer

The limit does not exist.

Question 17

Suppose $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 9}{x - 3}. $$What is $$\lim\limits_{x 	o 3} f(x)$$?

Accepted Answer

$$6$$

Question 18

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

Accepted Answer

$$\lim\limits_{x 	o a^+} f(x) is $$the limit as $$x$$ approaches $$a$$ from the right.

Definitions and Evaluation of Limits in Calculus

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