Precalculus Introduction to Limits Worksheet (PDF)

Precalculus math worksheet: Introduction to Limits – Maths PDF

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This limits practice worksheet for Precalculus covers introduction to limits (Common Core F-IF.B.6, F-IF.C.7d), with a free PDF to print.

These 20 problems introduce limits (Common Core F-IF.B.6 and F-IF.C.7d). You will find average rates of change, estimate limits from tables and graphs, evaluate limits algebraically, use one-sided limits and limits at infinity and test continuity.

Precalculus · Common Core State Standards: F-IF.B.6, F-IF.C.7d

Limits practice worksheet: exercises 1 to 10

Exercise 1 – Average rate of change

Find the average rate of change of $f(x)=x^{2}$ from $x=1$ to $x=3$.

Exercise 2 – Average rate of a cubic

Find the average rate of change of $f(x)=x^{3}$ from $x=1$ to $x=2$.

Exercise 3 – Average velocity

A rock falls so that $s(t)=16t^{2}$ feet after $t$ seconds. Find the average velocity from $t=1$ to $t=3$.

Exercise 4 – Shrinking interval

Secant lines from P.

For $f(x)=x^{2}$, find the average rate of change on $[2,2+h]$ and describe what happens as $h\to0$.

Exercise 5 – Limit from a table

A graph with a hole.

Complete the table for $f(x)=\frac{x^{2}-9}{x-3}$ at $x=2.9,\ 2.99,\ 3.01,\ 3.1$ and estimate $\lim_{x\to3}f(x)$.

Exercise 6 – Direct substitution

Evaluate $\lim_{x\to2}(3x^{2}-x+1)$.

Exercise 7 – Factor and cancel

Evaluate $\lim_{x\to-1}\frac{x^{2}-1}{x+1}$.

Exercise 8 – Rationalize

Evaluate $\lim_{x\to4}\frac{\sqrt{x}-2}{x-4}$.

Exercise 9 – Expand first

Evaluate $\lim_{x\to0}\frac{(x+5)^{2}-25}{x}$.

Exercise 10 – One-sided limits from a graph

A graph with a jump at x = 2.

Use the graph to find $\lim_{x\to2^{-}}f(x)$, $\lim_{x\to2^{+}}f(x)$, $f(2)$ and say whether $\lim_{x\to2}f(x)$ exists.

Exercises 11 and up

Exercise 11 – A piecewise limit

Let $f(x)=2x-1$ for $x< 3$ and $f(x)=x+2$ for $x\ge3$. Find the one-sided limits at $x=3$ and decide if $f$ is continuous at $3$.

Exercise 12 – Limit at infinity (equal degrees)

Graph with a horizontal asymptote.

Evaluate $\lim_{x\to\infty}\frac{2x+1}{x-1}$.

Exercise 13 – Limit at infinity (higher powers)

Evaluate $\lim_{x\to\infty}\frac{3x^{2}-5x}{x^{2}+4}$.

Exercise 14 – Limit at infinity (lower numerator)

Evaluate $\lim_{x\to\infty}\frac{4x+1}{x^{2}+2}$.

Exercise 15 – Infinite one-sided limits

Describe $\lim_{x\to2^{+}}\frac{1}{x-2}$ and $\lim_{x\to2^{-}}\frac{1}{x-2}$.

Exercise 16 – Removable discontinuity

Graph of f.

Let $f(x)=\frac{x^{2}-4}{x-2}$ for $x\neq2$ and $f(2)=5$. Is $f$ continuous at $2$? How can you remove the discontinuity?

Exercise 17 – Make a function continuous

Find $k$ so that $f(x)=x^{2}+1$ for $x< 2$ and $f(x)=kx-1$ for $x\ge2$ is continuous at $2$.

Exercise 18 – Delivery fee

Delivery fee by weight.

A store charges $\$5$ to ship up to $1$ lb, $\$8$ for over $1$ lb up to $2$ lb, and $\$11$ for over $2$ lb up to $3$ lb. Find the left and right limits of the fee $C(w)$ at $w=1$.

Exercise 19 – Instantaneous velocity

A particle’s position is $s(t)=t^{2}+2t$ feet. Find the average velocity on $[3,3+h]$ and the instantaneous velocity at $t=3$.

Exercise 20 – Drug concentration

Concentration over time.

The concentration of a drug is $C(t)=\frac{5t}{t+2}$ mg/L after $t$ hours. Find $C(8)$ and $\lim_{t\to\infty}C(t)$, and explain what the limit means.

Key facts behind this limits practice worksheet

Key facts

Average rate of change: (f(b) – f(a))/(b – a), the secant slope.
lim x to a of f(x) = L means f(x) approaches L as x approaches a from both sides.
The limit can exist even if f(a) is undefined (a hole).
Two-sided limit exists only if the left and right limits are equal.
f is continuous at a if f(a) exists, the limit exists and they are equal.

How to use this limits practice 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 practice worksheet as a PDF with the button below and keep it with your notes.

All Precalculus chapters: Precalculus math.

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