Q: If $$\lim_{x \to 5} f(x) = 7$$ and $$f(5) = 3$$, what can you say about the continuity of $$f(x) at$$ $$x = 5$$?

$$f(x) is $$not continuous at $$x = 5$$ because $$\lim_{x \to 5} f(x) \neq f(5).$$

Q: A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 9}{x - 3}$$ for $$x \neq 3$$ and $$f(3) = 7. Is$$ $$f(x)$$ continuous at $$x = 3$$?

No, because $$\lim_{x \to 3} f(x) \neq f(3).$$

Question 1

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

Accepted Answer

The value that $$f(x)$$ approaches as $$x$$ gets arbitrarily close to $$a.$$

Question 2

Evaluate $$\lim_{x 	o 2} (x^2$ - 4).$$

Accepted Answer

$$0$$

Question 3

A function $$f(x) is $$said to be continuous at $$x = a if$$:

Accepted Answer

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

Question 4

Which of the following is NOT a reason why a limit might not exist at $$x = a$$?

Accepted Answer

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

Question 5

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

Accepted Answer

$$6$$

Question 6

If $$\lim_{x 	o 1^-} f(x) = 2$$ and $$\lim_{x 	o 1^+} f(x) = 5$$, what can you conclude about $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

The limit does not exist.

Question 7

A function $$f(x) is $$defined as $$f(x) = \sqrt{x-2}. $$For which values of $$x is$$ $$f(x)$$ continuous?

Accepted Answer

$$[2, \infty)$$

Question 8

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

Accepted Answer

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

Question 9

Suppose $$f(x) is $$continuous everywhere except at $$x = 4$$, where $$\lim_{x 	o 4^-} f(x) = 2$$ and $$\lim_{x 	o 4^+} f(x) = -1. $$Which type of discontinuity does $$f(x)$$ have at $$x = 4$$?

Accepted Answer

Jump discontinuity

Question 10

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 11

If $$f(x) is a $$rational function and the denominator is not zero at $$x = a$$, what is $$\lim_{x 	o a} f(x)$$?

Accepted Answer

$$f(a)$$

Question 12

Which of the following real-world scenarios best illustrates a jump discontinuity?

Accepted Answer

A train ticket price that increases suddenly at age 18.

Question 13

Evaluate $$\lim_{x 	o 2} \frac{x^2$ - 4}{x^2$ + 2x - 8}.$$

Accepted Answer

$$-1$$

Question 14

Which of the following statements is true about the function $$f(x) = \frac{1}{x}$$?

Accepted Answer

$$f(x) is $$not continuous at $$x = 0$$ because the limit does not exist there.

Question 15

If $$\lim_{x 	o 5} f(x) = 7$$ and $$f(5) = 3$$, what can you say about the continuity of $$f(x) at$$ $$x = 5$$?

Accepted Answer

$$f(x) is $$not continuous at $$x = 5$$ because $$\lim_{x 	o 5} f(x) 
eq f(5).$$

Question 16

Evaluate $$\lim_{x 	o 1} \frac{x^2$ - 1}{x - 1}.$$

Accepted Answer

$$2$$

Question 17

Which of the following is an example of an infinite discontinuity?

Accepted Answer

$$f(x) = \frac{1}{x-2} at$$ $$x = 2$$

Question 18

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 9}{x - 3}$$ for $$x 
eq 3$$ and $$f(3) = 7. Is$$ $$f(x)$$ continuous at $$x = 3$$?

Accepted Answer

No, because $$\lim_{x 	o 3} f(x) 
eq f(3).$$

Question 19

Evaluate $$\lim_{x 	o 0} \frac{x^2$}{x}.$$

Accepted Answer

$$0$$

Question 20

Which of the following best describes the behavior of $$f(x) = 	an x as$$ $$x$$ approaches $$\frac{\pi}{2}$$ from the left?

Accepted Answer

$$f(x)$$ increases without bound (approaches $$+\infty).$$

Study Guide: Limits and Continuity (Calculus)

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