Q: A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 25}{x + 5}$$ for $$x \neq -5$$ and $$f(-5) = 1. Is$$ $$f(x)$$ continuous at $$x = -5$$?

No, because $$\lim_{x \to -5} f(x) \neq f(-5).$$

Q: Which of the following is a necessary condition for a function $$f(x) to be $$continuous at $$x = a$$?

All of the above, plus $$\lim_{x \to a} f(x) = f(a).$$

Question 1

Which of the following statements best describes the definition of continuity at a point $$a$$ for a function $$f(x)$$?

Accepted Answer

A function $$f is $$continuous at $$a if$$ $$f(a) is $$defined, $$\lim_{x 	o a} f(x)$$ exists, and $$\lim_{x 	o a} f(x) = f(a).$$

Question 2

Evaluate $$\lim_{x 	o 2} \frac{x^2$ + 3x - 10}{x^2$ - 4}$$ using algebraic methods.

Accepted Answer

Question 3

A company models its profit with $$f(x) = \frac{x^2$ - 2x + 1}{x - 1}. $$What is $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

$$2$$

Question 4

Which of the following functions is NOT continuous everywhere on $$(-\infty, \infty)$$?

Accepted Answer

$$F(x) = \sqrt{x - 8}$$

Question 5

Evaluate $$\lim_{x 	o \infty} \frac{x + 1}{x^2$ + 3x + 1}.$$

Accepted Answer

$$0$$

Question 6

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 25}{x + 5}$$ for $$x 
eq -5$$ and $$f(-5) = 1. Is$$ $$f(x)$$ continuous at $$x = -5$$?

Accepted Answer

No, because $$\lim_{x 	o -5} f(x) 
eq f(-5).$$

Question 7

Evaluate $$\lim_{x 	o 6} \frac{\sqrt{x + 10} - 4}{x - 6}.$$

Accepted Answer

$$\frac{1}{8}$$

Question 8

Which of the following is a necessary condition for a function $$f(x) to be $$continuous at $$x = a$$?

Accepted Answer

All of the above, plus $$\lim_{x 	o a} f(x) = f(a).$$

Question 9

Evaluate $$\lim_{x 	o 0} \frac{x^2$}{x^2$ - 4x}.$$

Accepted Answer

$$0$$

Question 10

A function $$f(x) is $$defined as $$f(x) = \frac{1}{3x + 4}$$ for $$x \leq 3$$ and $$f(x) = 2x - 1$$ for $$x \u003e 3. Is$$ $$f(x)$$ continuous at $$x = 3$$?

Accepted Answer

No, because the left and right limits at $$x = 3$$ are not equal.

Question 11

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

Accepted Answer

$$1$$

Question 12

Which of the following functions has a discontinuity at $$x = 2$$?

Accepted Answer

$$g(x) = \frac{x + 5}{x^2$ - 3x + 2}$$

Question 13

Evaluate $$\lim_{x 	o 2} \frac{\sqrt{x + 7} - 3}{x - 2}.$$

Accepted Answer

$$\frac{1}{6}$$

Question 14

A function $$h(x) is $$defined as $$h(x) = x - 1$$ for $$x  1. At $$which value(s) of $$x is$$ $$h(x)$$ potentially discontinuous?

Accepted Answer

$$x = -1$$ and $$x = 1$$

Question 15

Evaluate $$\lim_{x 	o 10} \frac{x^2$ - 17x + 70}{x - 10}.$$

Accepted Answer

Question 16

Which of the following is true about $$\lim_{x 	o \infty} \frac{c}{x^n}$$, where $$c is a $$constant and $$n is a $$positive integer?

Accepted Answer

The limit is $$0.$$

Question 17

Evaluate $$\lim_{x 	o 0} \sqrt{2x + 1}.$$

Accepted Answer

$$1$$

Question 18

A function $$F(x) = \sqrt{x^2$ - 4} is $$continuous for which values of $$x$$?

Accepted Answer

For $$x \leq -2$$ and $$x \geq 2$$

Question 19

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

Accepted Answer

$$\infty$$

Limits and Continuity: Business Calculus Study Notes

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