Q: Which of the following is NOT a valid reason for a function $$f(x) to be $$discontinuous at $$x = a$$?

$$\lim_{x \to a} f(x)$$ exists and equals $$f(a)$$

Q: For what value of $$a is $$the function $$f(x) = \frac{x^2$ - 9}{x - 3}$$ continuous at $$x = 3$$?

$$a = 6 ($$since $$\lim_{x \to 3} f(x) = 6$$ and $$f(3) = a)$$

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

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

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

$$f(x) is $$continuous for all $$x \neq 0.$$

Q: If $$f(x) is $$continuous at $$x = a$$, which of the following must be true?

$$\lim_{x \to a} f(x)$$ exists and equals $$f(a)$$

Q: Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

$$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

A rock is launched vertically upward from the ground at a speed of 96 ft/sec. Its position after $$t$$ seconds is given by $$s(t) = 96t - 16t^2$. What is the instantaneous velocity of the rock at t = 2$ seconds?

Accepted Answer

$$32 ft/sec$$

Question 2

Evaluate $$\lim_{x 	o 2} (x^2$ - 2x)$$ using the definition of a limit.

Accepted Answer

$$0$$

Question 3

Which of the following statements best describes the difference between the average velocity and the instantaneous velocity of an object?

Accepted Answer

Average velocity is the slope of the secant line over an interval, while instantaneous velocity is the slope of the tangent line at a point.

Question 4

Given $$\lim_{x 	o 1} f(x) = 4$$, $$\lim_{x 	o 1} g(x) = 5$$, and $$\lim_{x 	o 1} h(x) = 8$$, compute $$\lim_{x 	o 1} [f(x)g(x) + h(x)].$$

Accepted Answer

$$28$$

Question 5

Which of the following is NOT a valid reason for a function $$f(x) to be $$discontinuous at $$x = a$$?

Accepted Answer

$$\lim_{x 	o a} f(x)$$ exists and equals $$f(a)$$

Question 6

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

Accepted Answer

$$1$$

Question 7

Suppose $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 4}{x - 2}. $$What is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 8

Which of the following best describes an infinite limit?

Accepted Answer

The function increases or decreases without bound as $$x$$ approaches $$a.$$

Question 9

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

Accepted Answer

$$2$$

Question 10

For what value of $$a is $$the function $$f(x) = \frac{x^2$ - 9}{x - 3}$$ continuous at $$x = 3$$?

Accepted Answer

$$a = 6 ($$since $$\lim_{x 	o 3} f(x) = 6$$ and $$f(3) = a)$$

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

$$f(x) is $$continuous for all $$x 
eq 0.$$

Question 14

Find $$\lim_{x 	o 0^+} \ln x.$$

Accepted Answer

$$-\infty$$

Question 15

If $$f(x) is $$continuous at $$x = a$$, which of the following must be true?

Accepted Answer

$$\lim_{x 	o a} f(x)$$ exists and equals $$f(a)$$

Question 16

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

Accepted Answer

$$1$$

Question 17

Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 18

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x \u003c 1 \ 2x + 1 & x \geq 1 \end{cases}. $$For which value(s) of $$x is$$ $$f(x)$$ not continuous?

Accepted Answer

$$x = 1$$

Question 19

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

Accepted Answer

$$1$$

Limits and the Foundations of Calculus: Study Notes

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드