Q: Which of the following statements about continuity at a point $$x = a is $$true?

A function $$f is $$continuous at $$a if$$ $$\lim_{x \to a} f(x)$$ exists and equals $$f(a).$$

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

$$\lim_{x \to 0^+} f(x) = \infty$$

Q: Which of the following is a correct application of the Squeeze Theorem?

If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$$, then $$\lim_{x \to a} f(x) = L.$$

Question 1

Which of the following is the correct definition of the limit of a function $$f(x) as$$ $$x$$ approaches $$a$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists a $$\delta \u003e 0$$ such that if $$0 \u003c |x - a| \u003c \delta$$, then $$|f(x) - L| \u003c \epsilon.$$

Question 2

Evaluate $$\lim_{x 	o 2} (3x^2$ + 4x - 5)$$ using the limit laws for polynomials.

Accepted Answer

$$11$$

Question 3

Suppose $$\lim_{x 	o a} f(x) = 4$$, $$\lim_{x 	o a} g(x) = 5$$, and $$\lim_{x 	o a} h(x) = 8. $$What is $$\lim_{x 	o a} (f(x)g(x) + h(x))$$?

Accepted Answer

$$28$$

Question 4

Which of the following statements about continuity at a point $$x = a is $$true?

Accepted Answer

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

Question 5

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x}$$ using the Squeeze Theorem.

Accepted Answer

$$1$$

Question 6

Given $$f(x) = \frac{2x^2$ + 4}{x - 1}$$, what is the limit as $$x 	o 1$$?

Accepted Answer

$$\infty$$

Question 7

Which of the following is NOT a requirement for a function $$f to be $$continuous at $$x = a$$?

Accepted Answer

$$f(x) is $$differentiable at $$x = a$$

Question 8

Find $$\lim_{x 	o \infty} \frac{5x^2$ + 3x + 1}{2x^2$ - x + 4}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 9

A hospital monitors the concentration of a drug in a patient’s bloodstream, modeled by $$C(t) = \frac{10t}{t^2$ + 1}. $$What is $$\lim_{t 	o \infty} C(t)$$?

Accepted Answer

$$0$$

Question 10

Which theorem guarantees that a continuous function $$f on$$ $$[a, b]$$ takes every value between $$f(a)$$ and $$f(b)$$?

Accepted Answer

Intermediate Value Theorem

Question 11

Evaluate $$\lim_{x 	o 0} \frac{1 - \cos x}{x^2$}.$$

Accepted Answer

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

Question 12

Which of the following functions is continuous everywhere on $$\mathbb{R}$$?

Accepted Answer

$$f(x) = \sin x$$

Question 13

Suppose $$f(x) is $$continuous on $$[1, 5]$$ and $$f(1) = 2$$, $$f(5) = 8. $$Which value must $$f(x)$$ take for some $$x in$$ $$[1, 5]$$?

Accepted Answer

$$4$$

Question 14

Evaluate $$\lim_{x 	o \infty} \frac{3x^3$ - 2x + 1}{5x^3$ + 4x^2$ - 7}.$$

Accepted Answer

$$\frac{3}{5}$$

Question 15

A company’s profit function is $$P(x) = \frac{100x}{x + 10}. $$What is the horizontal asymptote of $$P(x) as$$ $$x 	o \infty$$?

Accepted Answer

$$y = 100$$

Question 16

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

Accepted Answer

$$\lim_{x 	o 0^+} f(x) = \infty$$

Question 17

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

Accepted Answer

$$1$$

Question 18

Which of the following is a correct application of the Squeeze Theorem?

Accepted Answer

If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x 	o a} g(x) = \lim_{x 	o a} h(x) = L$$, then $$\lim_{x 	o a} f(x) = L.$$

Question 19

Which of the following is true about the continuity of $$f(x) = \sqrt{x}$$?

Accepted Answer

$$f(x) is $$continuous for all $$x \geq 0$$

Calculus Study Notes: Limits and Continuity

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