Q: Suppose $$f(x) is $$defined on $$\mathbb{R}$$ and $$f(3) = -10. $$For $$f(x) to be $$continuous at $$x = 3$$, which of the following must be true?

$$\displaystyle \lim_{x \to 3} f(x) = -10$$

Q: Given $$f(x) = x + \frac{1}{x}$$ and $$g(x) = x + \frac{1}{x+2}$$, what is the domain of $$f \circ g$$?

$$\{ x \in \mathbb{R} \mid x \neq -2, x \neq -1 \}$$

Q: Which of the following is an example of a function that is continuous at all irrational points and discontinuous at all rational points?

The Thomae function: $$f(x) = \begin{cases} 1/q & x = p/q \text{ in lowest terms}, q \u003e 0 \\ 0 & x \notin \mathbb{Q} \end{cases}$$

Question 1

Which of the following is TRUE about the vertical line test for functions?

Accepted Answer

A graph in the $$x$$-$$y$$ plane represents a function if every vertical line intersects the graph at most once.

Question 2

Suppose $$f(x) is $$defined on $$\mathbb{R}$$ and $$f(3) = -10. $$For $$f(x) to be $$continuous at $$x = 3$$, which of the following must be true?

Accepted Answer

$$\displaystyle \lim_{x 	o 3} f(x) = -10$$

Question 3

Which theorem guarantees that the function $$f(x) = x^4$ + x - 3$$ has a root in the interval $$[1, 2]$$?

Accepted Answer

Intermediate Value Theorem

Question 4

Let $$f(x) = |x|. Is$$ $$f(x)$$ differentiable at $$x = 0$$?

Accepted Answer

No, because the left and right derivatives at $$x = 0$$ are not equal.

Question 5

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

Accepted Answer

$$1$$

Question 6

Given $$f(x) = x + \frac{1}{x}$$ and $$g(x) = x + \frac{1}{x+2}$$, what is the domain of $$f \circ g$$?

Accepted Answer

$$\{ x \in \mathbb{R} \mid x 
eq -2, x 
eq -1 \}$$

Question 7

For the piecewise function

$$f(x) = \begin{cases} x & x  2 \end{cases}$$

What is $$\displaystyle \lim_{x 	o 1^-} f(x)$$?

Accepted Answer

$$1$$

Question 8

Which of the following statements is TRUE regarding limits of functions?

Accepted Answer

A function $$f(x)$$ cannot have two different limits at the same point $$x = c.$$

Question 9

Evaluate $$\displaystyle \lim_{x 	o 0^-} \frac{|x|}{x}.$$

Accepted Answer

$$-1$$

Question 10

Suppose $$f(x) is $$defined as

$$f(x) = \begin{cases} 1 + x^2$ & x \leq 0 \ 2 - x & 0  2 \end{cases}

At $$which value(s) of $$x$$ does $$f(x)$$ fail to be continuous?

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 11

Which of the following is the correct $$\epsilon$$-$$\delta$$ definition of continuity for $$f(x) at$$ $$x = c$$?

Accepted Answer

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

Question 12

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

Accepted Answer

$$0$$

Question 13

Which of the following is an example of a function that is continuous at all irrational points and discontinuous at all rational points?

Accepted Answer

The Thomae function: $$f(x) = \begin{cases} 1/q & x = p/q 	ext{ in lowest terms}, q \u003e 0 \ 0 & x 
otin \mathbb{Q} \end{cases}$$

Question 14

Let $$f(x) = x^2$ - 1. $$Using any method except graphing, what is $$\displaystyle \lim_{x 	o 0} f(x)$$?

Accepted Answer

$$-1$$

Question 15

Suppose $$f(x) = x + \frac{1}{x}$$ and $$g(x) = x + \frac{1}{x+2}. $$What is $$f \circ g (x)$$?

Accepted Answer

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

Question 16

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{|x|}{x}.$$

Accepted Answer

Does not exist

Question 17

For which value(s) of $$c$$ does $$\displaystyle \lim_{x 	o c} x^4$ \cos(2x)$$ exist?

Accepted Answer

For all real numbers $$c$$

Question 18

Suppose you are modeling the temperature $$T (in$$ $$^ C) of a $$chemical reaction as a function of time $$t (in $$minutes) by $$T(t) = 2t + 5. $$Using the $$\epsilon$$-$$\delta$$ definition, which $$\delta$$ corresponds to a given $$\epsilon$$ for continuity at $$t = 3$$?

Accepted Answer

$$\delta = \frac{\epsilon}{2}$$

Calculus I (Limits, Continuity, and Functions) Guided Study Notes

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