A hole in the graph where $$\lim_{x \to a} f(x)$$ exists but $$f(a) is $$not defined or $$f(a) \neq \lim_{x \to a} f(x).$$

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

Question 1

Which of the following best describes the interval $$[a, b)$$ for real numbers $$a \u003c b$$?

Accepted Answer

$$[a, b) = \{ x \in \mathbb{R} : a \leq x \u003c b \}$$; it includes $$a$$ but not $$b.$$

Question 2

Let $$f(x) = |x|. $$What is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$0$$

Question 3

Suppose $$f(x) = \frac{\sin(x)}{x}. $$What is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$1$$

Question 4

Given the piecewise function $$f(x) = \begin{cases} x-1 & x \geq 0 \ x+1 & x \u003c 0 \end{cases}$$, what is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

The limit does not exist.

Question 5

Which of the following statements about the existence of $$\lim_{x 	o a} f(x) is $$correct?

Accepted Answer

The limit exists if and only if both the left-hand and right-hand limits exist and are equal.

Question 6

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

Accepted Answer

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

Question 7

Let $$f(x) = \frac{1}{x-2}. $$Which of the following statements is true?

Accepted Answer

$$x = 2 is a $$vertical asymptote of $$f(x).$$

Question 8

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

Accepted Answer

$$2$$

Question 9

Which of the following is an example of a removable discontinuity?

Accepted Answer

A hole in the graph where $$\lim_{x 	o a} f(x)$$ exists but $$f(a) is $$not defined or $$f(a) 
eq \lim_{x 	o a} f(x).$$

Question 10

Let $$f(x) = \sqrt{x}. $$For which values of $$x is$$ $$f(x)$$ defined and continuous?

Accepted Answer

$$x \geq 0$$

Question 11

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b). $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

There exists $$c \in (a, b)$$ such that $$f(c) = 0.$$

Question 12

Let $$f(x) = x^2$. What is the average rate of change of f(x$$)$$ on the interval $$[1, 3]$$?

Accepted Answer

$$4$$

Question 13

The derivative of $$f(x) at$$ $$x = a is $$defined as:

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

For which value(s) of $$x is $$the function $$f(x) = |x|$$ not differentiable?

Accepted Answer

$$x = 0$$

Question 16

What is the derivative of $$f(x) = x^3$ + 2x^2$ + 1$$?

Accepted Answer

$$3x^2 + 4x$

Question 17

Which rule is used to differentiate $$f(x) = (x^2$ + 1)(x^3$ - 2)$$?

Accepted Answer

Product Rule

Question 18

If $$f(x) = \sin(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\cos(x)$$

Question 19

Let $$f(x) = \sqrt{3x+1}. $$Using the chain rule, what is $$f'(x)$$?

Accepted Answer

$$\frac{3}{2\sqrt{3x+1}}$$

Limits and Interval Notation: Foundations of Calculus

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