Q: Which of the following best describes a removable discontinuity for a function $$f(x) at$$ $$x = a$$?

The limit $$\displaystyle \lim_{x \to a} f(x)$$ exists, but $$f(a) is $$either not defined or $$f(a) \neq \lim_{x \to a} f(x).$$

Q: Suppose $$f(x)$$ has a jump discontinuity at $$x = 0. $$Which of the following must be true?

$$\displaystyle \lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x)$$

Question 1

Which of the following best describes a removable discontinuity for a function $$f(x) at$$ $$x = a$$?

Accepted Answer

The limit $$\displaystyle \lim_{x 	o a} f(x)$$ exists, but $$f(a) is $$either not defined or $$f(a) 
eq \lim_{x 	o a} f(x).$$

Question 2

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

Accepted Answer

$$0$$

Question 3

Suppose $$f(x)$$ has a jump discontinuity at $$x = 0. $$Which of the following must be true?

Accepted Answer

$$\displaystyle \lim_{x 	o 0^-} f(x) 
eq \lim_{x 	o 0^+} f(x)$$

Question 4

Given $$f(x) = \frac{2x^2$ - 3x + 1}{x^2$ - 4}$$, find all vertical asymptotes.

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 5

Let $$f(x) = \begin{cases} 3x^2$ - 13, & x \u003c 2 \ ax, & x \geq 2 \end{cases}. $$What value of $$a$$ makes $$f(x)$$ continuous at $$x = 2$$?

Accepted Answer

$$a = -\frac{7}{4}$$

Question 6

Which of the following is the correct statement of the $$\varepsilon$$-$$\delta$$ definition of $$\displaystyle \lim_{x 	o a} f(x) = L$$?

Accepted Answer

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

Question 7

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{\sin 2x}{5x}.$$

Accepted Answer

$$2/5$$

Question 8

A chemical reaction's temperature $$T (in$$ $$^\circC) at $$time $$t (in $$minutes) is modeled by $$T(t) = 2t^2$ - 3t + 5. $$What is the average rate of change of temperature from $$t = 1 to$$ $$t = 4$$?

Accepted Answer

$$9$$

Question 9

Use the Intermediate Value Theorem to justify that the equation $$x^3$$ - $$7x^2 + 9x - 6 = 0$ has a solution between x = 3$ and x = 4$.

Accepted Answer

The function is continuous on $$[3,4]$$ and changes sign between $$x = 3$$ and $$x = 4.$$

Question 10

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

Accepted Answer

$$-\infty$$

Question 11

Find the horizontal asymptote of $$f(x) = \frac{2x^2$ - 3}{x^2$ + 8}.$$

Accepted Answer

$$y = 2$$

Question 12

Suppose $$g(x) = 2x^2$ + 5$$ for $$x  0. $$What is $$\displaystyle \lim_{x 	o 0^-} g(x)$$?

Accepted Answer

$$0$$

Question 13

Which of the following best explains why $$g(x) = \cos(x^2$) is $$continuous everywhere?

Accepted Answer

Because $$\cos(x) is $$continuous everywhere and $$x^2 is $$continuous everywhere, so their composition is continuous everywhere.

Question 14

Evaluate $$\displaystyle \lim_{x 	o 3^+} \ln(x - 3).$$

Accepted Answer

$$-\infty$$

Question 15

Find the slope of the tangent line to $$f(x) = x^2$ - 2x + 3 at$$ $$x = -1$$ using the limit definition.

Accepted Answer

$$-5$$

Question 16

A patient’s blood concentration of a drug is modeled by $$C(t) = 4e$$^{-0.5t}$$, where t$ is in hours. What is $$\displaystyle \lim_{t 	o \infty} C(t)$$?

Accepted Answer

$$0$$

Question 17

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{1 - \cos 3x}{x}.$$

Accepted Answer

$$0$$

Question 18

Which of the following is true about the function $$f(x) = \frac{2x^2$ - 3x + 1}{x^2$ - 4} as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o 2$$

Question 19

Suppose $$h(x) = \frac{x^2$ - 9}{x - 3}. $$Define $$h(3) so $$that $$h(x) is $$continuous at $$x = 3.$$

Accepted Answer

$$h(3) = 6$$

Calculus I: Limits, Continuity, and Rates of Change – Guided Review

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