Q: Which of the following limits requires the use of L'Hospital's Rule to evaluate?

$$\lim_{x \to 0} \frac{e^x - 1}{x}$$

Question 1

Evaluate the following limit: $$\lim_{x 	o 0} \frac{\sin(7x)}{x}.$$

Accepted Answer

$$7$$

Question 2

In the theory of Special Relativity, the Lorentz contraction formula is $$L = L_0$ \sqrt{1 - \frac{v^2$}{c^2$}}. If a $$spaceship travels at $$0.8c ($$where $$c is $$the speed of light), what is the observed length $$L if $$the proper length $$L_0 is$$ $$100$$ meters?

Accepted Answer

$$60$$ meters

Question 3

Find $$\lim_{x 	o 0^+} (\sqrt{x} - \sqrt{2x}).$$

Accepted Answer

$$0$$

Question 4

Consider the function $$f(x) = \sqrt{x^2$ - 4}. $$What is the domain of $$f(x)$$?

Accepted Answer

$$(-\infty, -2] \cup [2, \infty)$$

Question 5

For $$f(x) = \sqrt{x^2$ - 4}$$, at which points does $$f(x)$$ have a vertical asymptote?

Accepted Answer

None

Question 6

Which of the following is NOT a valid application of the Intermediate Value Theorem?

Accepted Answer

A function that is discontinuous on $$[a, b]$$

Question 7

Using the Intermediate Value Theorem, show that the equation $$x^5 - x + 1 = 0$ has a solution in the interval (-2$$, 0)$$.

Accepted Answer

Because $$f(-2)  0$$, and $$f(x) is $$continuous, so there is a root.

Question 8

A hiker starts climbing a mountain at 7:00 AM and reaches the top at 1:00 PM. The next day, the hiker descends from the top at 7:00 AM and reaches the base at 1:00 PM. Which theorem guarantees that there is a time when the hiker is at the same spot on the path on both days?

Accepted Answer

Fixed Point Theorem

Question 9

Let $$f(x) be $$defined as $$f(x) = 0 if$$ $$x is $$irrational, $$f(x) = 1/q if$$ $$x = p/q in $$lowest terms, and $$f(0) = 1. $$What is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$0$$

Question 10

Which of the following statements about the function $$f(x) = \sqrt{x^2$ - 4} is $$true?

Accepted Answer

The function is continuous on its domain $$(-\infty, -2] \cup [2, \infty).$$

Question 11

If $$\lim_{x 	o a} f(x) = L$$ and $$\lim_{x 	o a} g(x) = M$$, what is $$\lim_{x 	o a} [f(x) + g(x)]$$?

Accepted Answer

$$L + M$$

Question 12

Which of the following is a necessary condition for the Intermediate Value Theorem to apply to a function $$f on$$ $$[a, b]$$?

Accepted Answer

$$f is $$continuous on $$[a, b]$$

Question 13

Suppose $$f(x) is $$continuous on $$[1, 4]$$ and $$f(1) = -2$$, $$f(4) = 5. $$Which value must $$f(x)$$ take for some $$x in$$ $$[1, 4]$$?

Accepted Answer

$$0$$

Question 14

Which of the following limits requires the use of L'Hospital's Rule to evaluate?

Accepted Answer

$$\lim_{x 	o 0} \frac{e^x - 1}{x}$$

Question 15

If $$f(x) is $$defined as $$f(x) = 0$$ for irrational $$x$$, $$f(x) = 1/q$$ for rational $$x = p/q in $$lowest terms, and $$f(0) = 1$$, which of the following describes the graph of $$f(x)$$ near $$x = 0$$?

Accepted Answer

It is discontinuous at $$x = 0.$$

MAC2311 Practice Test 1: Limits, Continuity, and Applications

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