How to Find Limits at Infinity: a Free Self-Checking Calculus Activity

Want the free Limits at Infinity activity? Click here

How to find limits at infinity comes down to one question: which part of the function wins when x gets huge? Here is the activity I use to practice it with my own Calculus students, and it is free: a self-checking mystery pixel art activity for Google Sheets with 10 limits, a recording sheet and an answer key.

Below is how I teach the method and how I run the activity. Type your email on the sign-up page and the PDF is yours right away.

How to find limits at infinity of rational functions

Divide every term by the highest power of x in the denominator. Every term with x left in its denominator goes to 0, and what is left is the limit. That gives three cases students can learn to spot:

Same degree on top and bottom: the limit is the ratio of the leading coefficients, so (4x2 - x) / (2x2 + 5) goes to 4/2 = 2. Bigger degree on the bottom: the limit is 0. Bigger degree on the top: there is no finite limit; the function goes to infinity or negative infinity, and the signs tell you which.

With a square root, remember that the square root of x2 is |x|. As x goes to negative infinity, |x| = -x, so 3x / √(x2 + 1) goes to -3, not 3. The limit at infinity is the function's horizontal asymptote, which is how AP Calculus connects the two in Topic 1.15.

Get the free activity and answer key

Before you teach

The activity lives in a Google Sheet. The directions page in the free PDF has the link: click it, choose Make a Copy, and save the copy to your Google Drive so you can find it later.

• Print the recording sheet, one per student, or have students work on scratch paper or whiteboards.
• You and your students need a Google account.
• Give each student their own copy, so nobody edits your original.

Assign the limits at infinity activity: one copy for each student

In Google Classroom it takes a minute. No Google Classroom works too: use the Sheet's Send button to email it, or copy its link and share it somewhere your students sign in, like their school email.

1. In Google Classroom, click Classwork, then Create, then Assignment.
2. Name it Limits at Infinity, add instructions if you want, and set a due date.
3. Click Add, choose Google Drive, and pick your copy of the Sheet.
4. In the dropdown, choose Make a copy for each student.
5. Click Assign.

The recording sheet: 10 limits at infinity

Students show their work on the recording sheet. The 10 limits cover all three cases (same degree, bigger on the bottom, bigger on the top), go to both infinity and negative infinity, and two have a square root in the denominator, where the sign is the whole point.

The mystery pixel picture: how the activity checks itself

Students type each answer into the Google Sheet. A correct answer fills in part of a hidden pixel picture, so students know right away that they got it, and the whole picture shows when all 10 are right.

Check the work with the answer key

The answer key is a copy of the recording sheet with every answer, for checking the paper work.

Here is the free Limits at Infinity activity

Type your name and email on the sign-up page. The PDF downloads on the next page, and a copy comes to your inbox.

• The cover and directions, with the link to the self-checking Google Sheet
• The recording sheet with 10 limits at infinity
• The answer key

Get the free activity

This activity is from the limits lessons in my Calculus course

• Finding Limits
• Finding Limits Algebraically
• Limits to Infinity
• Limits Review
• Continuity

See every Calculus lesson

Questions teachers ask about limits at infinity

How do you find limits at infinity of rational functions?

Compare the degrees. Same degree: the ratio of the leading coefficients. Bigger degree on the bottom: 0. Bigger degree on the top: the function goes to infinity or negative infinity.

How do you solve limits at infinity with square roots?

Divide by the highest power of x, and remember that the square root of x2 is |x|. As x goes to negative infinity, |x| is -x, which flips the sign of the answer.

How are limits at infinity and horizontal asymptotes connected?

If the limit as x goes to infinity (or negative infinity) is a number L, then y = L is a horizontal asymptote of the graph. AP Calculus teaches the two together in Topic 1.15.

Want every Calculus activity right now?

The membership has every course from 7th grade to Calculus: notes, activities, review games and tests, with answer keys. New lessons keep coming.

See plans and pricing

Get the free Limits at Infinity activity

The self-checking Google Sheet, the recording sheet and the answer key, in one PDF. Type your email and it is yours.

Get the Free Activity

Pin this for later