Question 1

Evaluate the following limit:

$$\lim_{x 	o \infty} \frac{x^2$ - 5}{7x^2$ + 4}$$

Accepted Answer

$$\frac{1}{7}$$

Question 2

Find the value of the following limit:

$$\lim_{x 	o \infty} \frac{x^2$ - 5}{x^2$ + 10}$$

Accepted Answer

$$1$$

Question 3

Given $$f(x) = \frac{9x^3$}{\sqrt{4x^2$ + 1}} - \sqrt{4x^2$ + 1}$$, which of the following describes the horizontal asymptote of $$f(x)$$?

Accepted Answer

There is no horizontal asymptote

Question 4

Evaluate the following limit:

$$\lim_{x 	o 0} \frac{3 - \cos x}{x^2$}$$

Accepted Answer

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

Question 5

For the function $$f(x)$$ defined as

$$\begin{cases}
\sqrt{x} & x \u003c 0 \
3 - x & 0 \leq x \u003c 3 \
(3 - x)^2$ & x \geq 3
\end{cases}

At $$which value(s) of $$x is$$ $$f(x)$$ discontinuous?

Accepted Answer

$$x = 0$$ and $$x = 3$$

Question 6

If $$f$$ and $$g$$ are continuous functions with $$f(13) = 6$$ and $$\lim_{x 	o 13} [2f(x) - g(x)] = 13$$, what is $$g(13)$$?

Accepted Answer

$$-1$$

Question 7

How would you define $$f(6) in $$order to make $$f(x)$$ continuous at $$x = 6$$ for $$f(x) = \frac{x^2$ - 3x - 18}{x - 6}$$?

Accepted Answer

$$f(6) = 9$$

Question 8

A car's displacement $$s(t) in $$meters is given by $$s(t) = 2t^2$ - t + 2$$, where $$t is $$measured in seconds. What is the instantaneous velocity of the car at $$t = 2$$ seconds?

Accepted Answer

$$6 m/s$$

Question 9

Which of the following statements is true about the vertical asymptotes of the function $$f(x) = \frac{9x^3$}{\sqrt{4x^2$ + 1}} - \sqrt{4x^2$ + 1}$$?

Accepted Answer

There are no vertical asymptotes

Question 10

Given $$f(x) = \frac{x^2$ - 3x - 18}{x - 6}$$, which of the following is the correct limit as $$x 	o 6$$?

Accepted Answer

$$9$$

Question 11

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

$$\begin{cases}
\sqrt{x} & x \u003c 0 \
3 - x & 0 \leq x \u003c 3 \
(3 - x)^2$ & x \geq 3
\end{cases}$$

Which of the following is the value of $$\lim_{x 	o 3^-} f(x)$$?

Accepted Answer

$$0$$

Question 12

If $$f(x) = \frac{3 - \cos x}{x^2$}$$, what is the value of $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

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

Question 13

A patient’s temperature $$T(t) in $$degrees Celsius is modeled by $$T(t) = 37 + 0.5t - 0.02t^2$, where t$ is hours after medication. What is the rate of change of temperature at t = 5$ hours?

Accepted Answer

$$0$$ °C/hour

Question 14

Which of the following is the correct definition of the instantaneous velocity of an object at time $$t$$?

Accepted Answer

The derivative of the displacement function at time $$t$$

Question 15

If $$\lim_{x 	o a} f(x) = L$$ and $$f(a) is $$defined, which of the following is necessary for $$f(x) to be $$continuous at $$x = a$$?

Accepted Answer

$$f(a) = L$$

Question 16

A chemical reaction’s concentration $$C(t) is $$given by $$C(t) = 5e$$^{-0.2t}$$, where t$ is time in minutes. What is the instantaneous rate of change of concentration at t = 0$?

Accepted Answer

$$-1$$ units/min

Question 17

Which of the following is the correct interpretation of $$\lim_{x 	o \infty} \frac{x^2$ - 5}{7x^2$ + 4} in $$the context of population growth, where $$x is $$time in years?

Accepted Answer

The population stabilizes at $$\frac{1}{7}$$ times the carrying capacity as time increases.

Limits, Continuity, and Asymptotes in Calculus: Study Notes

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