Q: Which of the following statements about continuity is TRUE?

A function is continuous at $$x = a if$$ $$\lim_{x \to a} f(x)$$ exists and equals $$f(a).$$

Q: Which of the following correctly expresses the limit definition of the derivative of $$f(x) at$$ $$x = a$$?

$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Q: A chemical reaction’s rate is modeled by $$R(t) = t^3$ - 2t. $$Find the slope of the tangent line at $$t = 1$$ using the limit definition.

$$f'(1) = \lim_{h \to 0} \frac{(1+h)^3$ - 2(1+h) - [1^3$ - 2 \cdot 1]}{h}$$

Q: Which of the following is an example of the Squeeze Theorem?

If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$$, then $$\lim_{x \to a} f(x) = L.$$

Q: A function $$f(x)$$ has a vertical asymptote at $$x = 3. $$Which limit notation best describes this?

$$\lim_{x \to 3} f(x) = \infty$$

Q: A function $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x \neq 2 \\ 5 & x = 2 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 2$$?

No, because $$\lim_{x \to 2} f(x) \neq f(2)$$

Question 1

Which of the following is a reason a limit $$\lim_{x 	o a} f(x)$$ might fail to exist?

Accepted Answer

The function approaches different values from the left and right at $$x = a.$$

Question 2

Estimate the limit $$\lim_{x 	o 0} \frac{\sin x}{x}$$ using a numerical approach. Which value is closest to the limit?

Accepted Answer

$$1$$

Question 3

Given the function $$f(x) = \frac{1}{x}$$, what is $$\lim_{x 	o 0^+} f(x)$$?

Accepted Answer

$$\infty$$

Question 4

Which of the following statements about continuity is TRUE?

Accepted Answer

A function is continuous at $$x = a if$$ $$\lim_{x 	o a} f(x)$$ exists and equals $$f(a).$$

Question 5

For the function $$f(x) = \frac{x^2$ - 1}{x - 1}$$, determine the type of discontinuity at $$x = 1.$$

Accepted Answer

Removable discontinuity

Question 6

Which theorem guarantees that a function $$f(x)$$ takes on every value between $$f(a)$$ and $$f(b)$$ for $$a \u003c b if$$ $$f is $$continuous on $$[a, b]$$?

Accepted Answer

Intermediate Value Theorem

Question 7

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

Accepted Answer

$$3$$

Question 8

A hospital monitors the concentration of a drug in a patient’s bloodstream, modeled by $$C(t) = \frac{10}{t+2}. $$What is the horizontal asymptote of $$C(t) as$$ $$t 	o \infty$$?

Accepted Answer

$$y = 0$$

Question 9

Which of the following correctly expresses the limit definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f'(a) = \lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$$

Question 10

If $$f(x) is $$differentiable at $$x = a$$, which of the following must also be true?

Accepted Answer

$$f(x) is $$continuous at $$x = a$$

Question 11

A chemical reaction’s rate is modeled by $$R(t) = t^3$ - 2t. $$Find the slope of the tangent line at $$t = 1$$ using the limit definition.

Accepted Answer

$$f'(1) = \lim_{h 	o 0} \frac{(1+h)^3$ - 2(1+h) - [1^3$ - 2 \cdot 1]}{h}$$

Question 12

Suppose $$f(x) is $$continuous everywhere except at $$x = 2$$, where $$f(2) is $$undefined but $$\lim_{x 	o 2} f(x)$$ exists. What type of discontinuity is this?

Accepted Answer

Removable discontinuity

Question 13

A population model $$P(t) = \frac{100}{t} is $$used for $$t \u003e 0. $$What is $$\lim_{t 	o 0^+} P(t)$$?

Accepted Answer

$$\infty$$

Question 14

Which of the following is an example of the Squeeze Theorem?

Accepted Answer

If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x 	o a} g(x) = \lim_{x 	o a} h(x) = L$$, then $$\lim_{x 	o a} f(x) = L.$$

Question 15

A function $$f(x)$$ has a vertical asymptote at $$x = 3. $$Which limit notation best describes this?

Accepted Answer

$$\lim_{x 	o 3} f(x) = \infty$$

Question 16

A function $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x 
eq 2 \ 5 & x = 2 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 2$$?

Accepted Answer

No, because $$\lim_{x 	o 2} f(x) 
eq f(2)$$

Question 17

A car’s position is given by $$s(t) = 4t^2$. What is the equation of the tangent line at t = 3$ using the limit definition of the derivative?

Accepted Answer

$$y = 24(t - 3) + 36$$

Question 18

Which of the following best describes the graphical interpretation of the Epsilon-Delta Definition of a Limit?

Accepted Answer

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

Calculus I Exam 1 Review: Limits, Continuity, and the Definition of the Derivative

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