Q: Which of the following is NOT a requirement for a function $f(x)$ to be continuous at $x=a$?

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

Q: Suppose $f(x)$ has a jump discontinuity at $x=1$. Which of the following is TRUE?

$\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x)$

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

$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Question 1

Evaluate the following limit analytically: $\lim_{x 	o 0} \frac{\sqrt{x+3} - \sqrt{3}}{x}$. What is the value of the limit?

Accepted Answer

$\frac{1}{2\sqrt{3}}$

Question 2

Which of the following statements is TRUE about the function $f(x) = \frac{1}{(x-2)^2}$?

Accepted Answer

It is continuous everywhere except at $x=2$ where it has a non-removable discontinuity.

Question 3

Given $g(x) = \frac{x-2}{x^2+2x-6}$, at which $x$-value does $g(x)$ have a removable discontinuity?

Accepted Answer

$x=2$

Question 4

Find $\lim_{x 	o 3} \frac{x+1-2}{x-3}$.

Question 5

A medication's concentration in the bloodstream is modeled by $C(t) = \frac{5t}{t^2+1}$. What is $\lim_{t 	o \infty} C(t)$?

Accepted Answer

$0$

Question 6

Suppose $f(x)$ is defined as $f(x) = \begin{cases} x^2-11x-6 & x \geq -4 \ 2x & x < -4 \end{cases}$. What is $\lim_{x 	o -4^-} f(x)$?

Accepted Answer

$-8$

Question 7

Which of the following is NOT a requirement for a function $f(x)$ to be continuous at $x=a$?

Accepted Answer

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

Question 8

Find the slope of the tangent line to $f(x) = 3x^2 + x - 4$ at $x = -1$ using the limit definition of derivative.

Accepted Answer

$-5$

Question 9

Given the graph of $f(x)$ below, which of the following $x$-values is a point of non-removable discontinuity?

Accepted Answer

$x=2$ (vertical asymptote)

Question 10

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

Accepted Answer

$1$

Question 11

A car's position is given by $s(t) = -16t^2 + 100t$. What is the average velocity over the interval $[3,5]$?

Accepted Answer

$-20$

Question 12

Which of the following limits does NOT exist?

Accepted Answer

$\lim_{x 	o 0} \frac{1}{x}$

Question 13

For $f(x) = \frac{x^2-13x+30}{x^2-100x}$, which $x$-values are holes (removable discontinuities)?

Accepted Answer

$x=10$

Question 14

Find $\lim_{x 	o 0} \frac{4(\sqrt{x+2})-2}{x}$.

Accepted Answer

$1$

Question 15

Which of the following is the correct equation for the tangent line to $y = x^2 - 9$ at the point $(2, -5)$?

Accepted Answer

$y = 4x - 13$

Question 16

Suppose $f(x)$ has a jump discontinuity at $x=1$. Which of the following is TRUE?

Accepted Answer

$\lim_{x 	o 1^-} f(x) 
eq \lim_{x 	o 1^+} f(x)$

Question 17

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

Accepted Answer

$1$

Question 18

A function $f(x)$ is continuous everywhere except at $x=3$, where $\lim_{x 	o 3} f(x)$ exists but $f(3)$ is not defined. What type of discontinuity is this?

Accepted Answer

Removable discontinuity

Question 19

Find $\lim_{x 	o 0} \frac{\cos x - 1}{x}$.

Accepted Answer

$0$

Question 20

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

Accepted Answer

$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$

Limits and Continuity: Study Guide for Calculus I

Study Guide - Practice Questions

Study Guide - Flashcards