Question 1

Let $$f(x) be $$defined on an open interval containing $$c$$, except possibly at $$c$$ itself. Which of the following best describes the meaning of $$\lim_{x 	o c} f(x) = L$$?

Accepted Answer

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

Question 2

Which of the following statements is NOT a requirement for the existence of $$\lim_{x 	o c} f(x) = L$$?

Accepted Answer

$$f(x)$$ must be defined at $$x = c.$$

Question 3

Suppose $$f(x) is $$defined as follows: $$f(x) = x^2$ for x$$ 
eq 2$$, and f(2$$)$$ is undefined. What is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 4

Given the formal definition of a limit, which of the following best describes the role of $$\varepsilon$$ and $$\delta$$?

Accepted Answer

$$\varepsilon$$ represents how close $$f(x) is to$$ $$L$$, and $$\delta$$ represents how close $$x is to$$ $$c.$$

Question 5

Consider the function $$f(x) = \frac{x^2$ - 9}{x - 3}. $$What is $$\lim_{x 	o 3} f(x)$$?

Accepted Answer

$$6$$

Question 6

A chemical reaction's rate $$R(t) is $$modeled by $$R(t) = \frac{10t}{t^2$ + 1}. $$What is $$\lim_{t 	o 0} R(t)$$?

Accepted Answer

$$0$$

Question 7

If $$\lim_{x 	o 2} f(x) = 5$$ and $$\lim_{x 	o 2} g(x) = -3$$, what is $$\lim_{x 	o 2} [2f(x) - 3g(x)]$$?

Accepted Answer

$$19$$

Question 8

A patient’s body temperature $$T(h) in $$degrees Celsius after taking medication is modeled by $$T(h) = 37 + 2e$$^{-h}$$. What is $$\lim_{h 	o \infty} T(h)$$?

Accepted Answer

$$37$$

Question 9

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

Accepted Answer

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

Question 10

Suppose $$f(x) is $$defined on $$(-1, 1)$$ except at $$x = 0$$, and $$\lim_{x 	o 0} f(x) = 7. $$Which of the following statements is true?

Accepted Answer

For every $$\varepsilon \u003e 0$$, there exists a $$\delta \u003e 0$$ such that if $$0 \u003c |x| \u003c \delta$$, then $$|f(x) - 7| \u003c \varepsilon.$$

Question 11

A population $$P(t)$$ grows according to $$P(t) = 1000 + 500e$$^{-0.1t}$$. What is $$\lim_{t 	o \infty} P(t)$$?

Accepted Answer

$$1000$$

Question 12

Which of the following is a correct interpretation of the statement $$\lim_{x 	o c} f(x) = L$$?

Accepted Answer

As $$x$$ gets arbitrarily close to $$c ($$but not equal to $$c$$), $$f(x)$$ gets arbitrarily close to $$L.$$

Question 13

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

Accepted Answer

$$1$$

Question 14

A medication’s concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-2t}$$, where t$ is in hours. What is $$\lim_{t 	o \infty} C(t)$$?

Accepted Answer

$$0$$

Question 15

If $$\lim_{x 	o 1} f(x) = 3$$ and $$\lim_{x 	o 1} g(x) = 2$$, what is $$\lim_{x 	o 1} [f(x)g(x)]$$?

Accepted Answer

$$6$$

Question 16

Suppose $$f(x) is $$defined as $$f(x) = \frac{1}{x}$$ for $$x 
eq 0. $$What is $$\lim_{x 	o 0^+} f(x)$$?

Accepted Answer

$$+\infty$$

Question 17

A car’s velocity $$v(t) is $$given by $$v(t) = 60 - 5t. $$What is $$\lim_{t 	o 12} v(t)$$?

Accepted Answer

$$0$$

Question 18

Which of the following is a necessary condition for $$\lim_{x 	o c} f(x) = L to $$exist?

Accepted Answer

$$f(x)$$ approaches the same value $$L$$ from both the left and right as $$x$$ approaches $$c.$$

Question 19

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

Accepted Answer

$$1$$

Limits and Continuity: Formal Definition and Properties

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