Q: Suppose $$f(x) is $$defined as follows: $$ f(x) = \begin{cases} x^2$, & x \u003c 1 \\ 2x - 1, & x \geq 1 \end{cases} Is$$ $$f(x)$$ continuous at $$x = 1$$?

Yes, because $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1).$$

Q: Suppose $$f(x) is $$not continuous at $$x = 3. $$Which of the following could be a reason?

$$f(3) is $$defined, but $$\lim_{x \to 3} f(x)$$ does not exist.

Question 1

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

Accepted Answer

$$6$$

Question 2

Use the Squeeze Theorem to evaluate the following limit: 
$$
\lim_{x 	o 0} x^2$ \sin\left(\frac{1}{x}ight)
$$

Accepted Answer

$$0$$

Question 3

Find the vertical asymptotes of the function: 
$$
f(x) = \frac{2x}{x^2$ - 9}
$$

Accepted Answer

$$x = 3$$ and $$x = -3$$

Question 4

Determine the horizontal asymptote(s) of the function: 
$$
f(x) = \frac{5x^2$ + 1}{2x^2$ - 3}
$$

Accepted Answer

$$y = \frac{5}{2}$$

Question 5

Is the function $$f(x) = \frac{x^2$ - 4}{x - 2}$$ continuous at $$x = 2$$? Use the continuity checklist.

Accepted Answer

No, because $$f(2) is $$not defined.

Question 6

Suppose $$f(x) is $$continuous on $$[1, 4]$$, $$f(1) = -2$$, and $$f(4) = 5. By $$the Intermediate Value Theorem, which of the following must be true?

Accepted Answer

There is a $$c in$$ $$(1, 4)$$ such that $$f(c) = 0.$$

Question 7

Use the epsilon-delta definition to prove that 
$$
\lim_{x 	o 1} (2x + 3) = 5
.
$$Which of the following is the correct choice for $$\delta in $$terms of $$\epsilon$$?

Accepted Answer

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

Question 8

Which of the following statements is true about the function $$f(x) = \frac{1}{x^2$} as$$ $$x 	o 0$$?

Accepted Answer

The limit is $$\infty.$$

Question 9

Given the graph of $$f(x)$$ has a sharp corner at $$x = 2$$, which of the following is true?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 2.$$

Question 10

Find the derivative of $$f(x) = x^2$ + 3x$$ using the limit definition of the derivative.

Accepted Answer

$$2x + 3$$

Question 11

Find the equation of the tangent line to $$f(x) = x^2$ at x = 1$.

Accepted Answer

$$y = 2x - 1$$

Question 12

Suppose $$f(x) is $$defined as follows:
$$
f(x) = \begin{cases} x^2$, & x \u003c 1 \ 2x - 1, & x \geq 1 \end{cases}

Is$$ $$f(x)$$ continuous at $$x = 1$$?

Accepted Answer

Yes, because $$\lim_{x 	o 1^-} f(x) = \lim_{x 	o 1^+} f(x) = f(1).$$

Question 13

Given the graph of $$f(x)$$, which of the following would indicate that $$f'(x)$$ does not exist at $$x = a$$?

Accepted Answer

A sharp corner at $$x = a.$$

Question 14

Evaluate the following limit using limit laws:
$$
\lim_{x 	o \infty} \frac{4x^3$ + 2}{2x^3$ - 5x}
$$

Accepted Answer

$$2$$

Question 15

A chemical reaction's rate is modeled by $$f(t) = \frac{10}{t+2}. $$What is the horizontal asymptote of $$f(t) as$$ $$t 	o \infty$$?

Accepted Answer

$$y = 0$$

Question 16

Suppose $$f(x) is $$not continuous at $$x = 3. $$Which of the following could be a reason?

Accepted Answer

$$f(3) is $$defined, but $$\lim_{x 	o 3} f(x)$$ does not exist.

Question 17

A patient’s body temperature $$T(t) is $$modeled by $$T(t) = 98 + 2e$$^{-0.5t}$$. What is $$\lim_{t 	o \infty} T(t)$$?

Accepted Answer

$$98$$

Question 18

Given $$f(x) = |x|$$, at which value(s) of $$x is$$ $$f(x)$$ not differentiable?

Accepted Answer

$$x = 0$$

Question 19

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 is a $$c in$$ $$(a, b)$$ such that $$f(c) = 0.$$

Limits, Continuity, and Derivatives: Study Guide

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