AP Calculus AB Introducing Limits Worksheet (PDF)

AP Calculus AB math worksheet: Introducing Limits – Maths PDF

Cover of the book AP Calculus AB Math Worksheets with Answers (PDF)

Download as a PDFThe AP Calculus AB worksheets book with answersPrintable PDFSee the book ›

Sommaire

This limits intro worksheet for AP Calculus AB covers introducing limits (Common Core AP CALC AB 1.1, AP CALC AB 1.2, AP CALC AB 1.3, AP CALC AB 1.4), with a free PDF to print.

These 22 problems practice estimating limits from tables and graphs, one-sided limits and the cases where a limit does not exist (AP Calculus AB, topics 1.1, 1.2, 1.3, 1.4). The early problems build the basic skill, and each one uses a different function or situation so that no two feel alike.

Later problems use piecewise functions, rates of change from tables, and real situations such as parking fees and a cooling coffee. Work in pencil, show every step, and check the answer key only after a full attempt.

AP Calculus AB · Common Core State Standards: AP CALC AB 1.1, AP CALC AB 1.2, AP CALC AB 1.3, AP CALC AB 1.4

Limits intro worksheet: exercises 1 to 10

Exercise 1 – Reading limit notation

Write $\lim_{x\to 5} f(x) = 12$ in words.

Exercise 2 – Table for a linear function

Complete a table for $f(x)=x+4$ at $2.9, 2.99, 2.999, 3.001, 3.01, 3.1$, then estimate $\lim_{x\to 3} f(x)$.

Exercise 3 – Table with an undefined point

Make a table for $g(x)=\frac{x^{2}-9}{x-3}$ using $2.9, 2.99, 2.999, 3.001, 3.01, 3.1$ and estimate $\lim_{x\to 3} \frac{x^{2}-9}{x-3}$.

Exercise 4 – Limit versus function value

The graph of f has an open circle at (2, 3) and a filled point at (2, 1).

Use the graph of $f$ to find $\lim_{x\to 2} f(x)$ and $f(2)$.

Exercise 5 – One-sided limits from a graph

A jump at x = 1: open circle at (1, 1), filled point at (1, 3).

Use the graph to find $\lim_{x\to 1^-} f(x)$, $\lim_{x\to 1^+} f(x)$ and $\lim_{x\to 1} f(x)$.

Exercise 6 – Limit at a smooth point

The graph of f(x) = x²/2 − 2 is a smooth parabola.

The graph shows $f(x)=\frac{x^{2}}{2}-2$. Find $\lim_{x\to 2} f(x)$.

Exercise 7 – Absolute value quotient

Estimate $\lim_{x\to 0} \frac{|x|}{x}$ using inputs $-0.1, -0.01, -0.001, 0.001, 0.01, 0.1$. Does the limit exist?

Exercise 8 – Unbounded outputs

Both branches of 1/x² climb without bound near x = 0.

The graph shows $f(x)=\frac{1}{x^{2}}$. Describe the behavior as $x\to 0$ and state whether the limit exists as a real number.

Exercise 9 – Sine over x

With the calculator in radian mode, evaluate $\frac{\sin x}{x}$ at $-0.1, -0.01, 0.01, 0.1$ and estimate $\lim_{x\to 0} \frac{\sin x}{x}$.

Exercise 10 – Square root difference

Estimate $\lim_{x\to 0} \frac{\sqrt{x+4}-2}{x}$ by evaluating the quotient at $-0.1, -0.01, -0.001, 0.001, 0.01, 0.1$.

Exercises 11 and up

Exercise 11 – Average rate of change

For $f(x)=x^{2}$, find the average rate of change on $[1,3]$, $[1,1.1]$ and $[1,1.01]$. Then estimate the instantaneous rate of change at $x=1$.

Exercise 12 – Limit from a table

Values of g near x = 5.

Use the table to estimate $\lim_{x\to 5} g(x)$ and to state $g(5)$.

Exercise 13 – Piecewise one-sided limits

Let $f(x)=x^{2}$ for $x < 1$ and $f(x)=2x+1$ for $x\ge 1$. Find the left-hand limit, the right-hand limit and the limit at $x=1$.

Exercise 14 – Making a limit exist

Let $f(x)=3x+1$ for $x < 2$ and $f(x)=x^{2}+k$ for $x\ge 2$. Find $k$ so that $\lim_{x\to 2} f(x)$ exists.

Exercise 15 – Square root near the edge

Estimate $\lim_{x\to 1^+} \sqrt{x-1}$ using $x=1.1, 1.01, 1.001$. Why does the left-hand limit at $x=1$ not make sense for this function?

Exercise 16 – Parking garage fee

A garage charges \$3 for each hour or part of an hour. The fee in dollars for parking $t$ hours is $C(t)$, with $C(t)=6$ for $1 < t\le 2$ and $C(t)=9$ for $2 < t\le 3$. Find the left-hand and right-hand limits of $C(t)$ as $t\to 2$ and decide whether the two-sided limit exists.

Exercise 17 – Speed of a skateboarder

Maya rolls down a ramp, and her distance from the top after $t$ seconds is $s(t)=t^{2}$ meters. Find her average speed over $[3,4]$, $[3,3.1]$, $[3,3.01]$ and $[3,3.001]$, then estimate her speed at $t=3$ seconds.

Exercise 18 – Cooling coffee

The temperature of a coffee in °F is $T$ at $t$ minutes: $T(0)=180$, $T(2)=150$, $T(4)=128$, $T(6)=110$, $T(8)=96$. Find the average rate of change on each 2-minute interval. What do the averages suggest about the rate near $t=8$?

Exercise 19 – Filling a water tank

A tank holds $V(t)=4t^{2}$ gallons after $t$ minutes. Find the average rate of change over $[5,5.1]$, $[5,5.01]$ and $[5,5.001]$, and estimate the instantaneous rate at $t=5$.

Exercise 20 – Hole in a rational graph

Let $g(x)=\frac{x^{2}-x-6}{x-3}$. (a) Explain why $g(3)$ is undefined. (b) Estimate $\lim_{x\to 3} g(x)$ with a table at $2.9, 2.99, 3.01, 3.1$. (c) Describe the graph at $x=3$.

Exercise 21 – Several limits from one graph

The graph of f has a jump at x = 1.

Use the graph of $f$ to find: (a) $\lim_{x\to 1^-} f(x)$; (b) $\lim_{x\to 1^+} f(x)$; (c) $f(1)$; (d) $\lim_{x\to 1} f(x)$; (e) $\lim_{x\to 3} f(x)$.

Exercise 22 – Does the limit equal the value?

Jamal says, “The limit of a function at $x=4$ is always $f(4)$.” Use $h(x)=\frac{x^{2}-16}{x-4}$ for $x\ne 4$ with $h(4)=2$ to show that he is wrong. Compute the limit using a table with $3.9, 3.99, 4.01, 4.1$.

Key facts behind this limits intro worksheet

Key facts

A limit describes nearby behavior, not the value at the point itself.
The two-sided limit exists only if the left-hand and right-hand limits are equal.
Tables and graphs give estimates; algebra gives exact values.
A limit fails to exist because of a jump, unbounded growth or oscillation.

How to use this limits intro worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the limits intro worksheet as a PDF with the button below and keep it with your notes.

All AP Calculus AB chapters: AP Calculus AB math.

Voter.. post

Free printable PDF