Q: Suppose $f(x)$ is continuous everywhere except at $x=1$ and $x=3$. Which of the following statements is true?

$f(x)$ is discontinuous at $x=1$ and $x=3$

Q: Which of the following best describes a removable discontinuity?

A point where the function is not defined but the limit exists

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

$\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Q: A function $g(x)$ is defined as $g(x) = \frac{1}{x}$ for $x \neq 0$. Which of the following statements is true?

$g(x)$ is discontinuous at $x=0$

Q: Suppose a function $h(x)$ has a jump discontinuity at $x=2$. Which of the following is true?

$\lim\limits_{x \to 2^-} h(x) \neq \lim\limits_{x \to 2^+} h(x)$

Question 1

Evaluate the following limit: $\lim\limits_{x 	o 2} (2x^2 - 3x + 6)$

Accepted Answer

$8$

Question 2

Find $\lim\limits_{x 	o 0} \left( \sin \left( \frac{5}{x-5} ight) ight)$.

Accepted Answer

$0$

Question 3

Evaluate $\lim\limits_{x 	o 0} \left( \frac{\frac{1}{x} - x}{x} ight)$.

Accepted Answer

$\infty$

Question 4

Given the piecewise function $f(x) = \begin{cases} 3x-2, & x<2 \ x^2, & 2 \leq x < 3 \ x-4, & 3 \leq x \end{cases}$, find $\lim\limits_{x 	o 3^-} f(x)$.

Accepted Answer

$9$

Question 5

Evaluate $\lim\limits_{x 	o 1} \left( \frac{x}{x^2 - x} ight)$.

Accepted Answer

$\infty$

Question 6

Evaluate $\lim\limits_{x 	o \infty} \left( \frac{x^2 - 2}{x^2 + x} ight)$.

Accepted Answer

$1$

Question 7

For which values of $x$ is the function $f(x) = \begin{cases} 3x-2, & x<2 \ x^2, & 2 \leq x < 3 \ x-4, & 3 \leq x \end{cases}$ continuous? Express your answer in interval notation.

Question 8

A function $f(x)$ is defined as $f(x) = \frac{x^2 - 4}{x-2}$. What is $\lim\limits_{x 	o 2} f(x)$?

Accepted Answer

$2$

Question 9

Suppose $f(x)$ is continuous everywhere except at $x=1$ and $x=3$. Which of the following statements is true?

Accepted Answer

$f(x)$ is discontinuous at $x=1$ and $x=3$

Question 10

Given $\lim\limits_{x 	o 0} \sqrt{x} \sin \left( \frac{1}{x} ight)$, which of the following is true?

Accepted Answer

The limit is $0$

Question 11

A car's position at time $t$ is given by $s(t) = t^2 + 2t$. What is the instantaneous velocity at $t=3$?

Accepted Answer

$10$

Question 12

Find the equation of the tangent line to $f(x) = x^2$ at the point $(2,4)$.

Accepted Answer

$y = 4x - 4$

Question 13

Which of the following best describes a removable discontinuity?

Accepted Answer

A point where the function is not defined but the limit exists

Question 14

If $\lim\limits_{x 	o a^-} f(x) = \lim\limits_{x 	o a^+} f(x) = L$ but $f(a) 
eq L$, what type of discontinuity does $f$ have at $x=a$?

Accepted Answer

Removable discontinuity

Question 15

A chemical reaction's concentration is modeled by $C(t) = 5e^{-0.2t}$. What is the instantaneous rate of change of concentration at $t=0$?

Accepted Answer

$-1$

Question 16

Which of the following is the definition of the derivative of $f(x)$ at $x=a$?

Accepted Answer

$\lim\limits_{h 	o 0} \frac{f(a+h) - f(a)}{h}$

Question 17

A function $g(x)$ is defined as $g(x) = \frac{1}{x}$ for $x 
eq 0$. Which of the following statements is true?

Accepted Answer

$g(x)$ is discontinuous at $x=0$

Question 18

Suppose a function $h(x)$ has a jump discontinuity at $x=2$. Which of the following is true?

Accepted Answer

$\lim\limits_{x 	o 2^-} h(x) 
eq \lim\limits_{x 	o 2^+} h(x)$

Question 19

A patient’s temperature $T(t)$ in degrees Celsius is modeled by $T(t) = 37 + 2e^{-0.5t}$. What is the rate of change of temperature at $t=0$?

Accepted Answer

$-1$

Calculus I: Limits, Continuity, and Tangent Lines – Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards