Math Bytes · Calculus · Episode 3 · Limits and continuity
Limits at infinity
A limit at infinity asks where a function's values head as x keeps growing. 1/x shrinks toward 0 without ever reaching it, so its limit as x approaches infinity is 0, and (2x + 1)/x = 2 + 1/x heads to 2.
Watch 1/x as x grows
Plug bigger and bigger numbers into 1/x:
x = 10:1/10 = 0.1x = 100:1/100 = 0.01x = 1000:1/1000 = 0.001
The answers keep shrinking toward 0. They never actually reach 0, because 1 divided by a positive number is always positive. But they get as close to 0 as you like, once x is big enough. That's exactly what we write as
lim x→∞ 1/x = 0
and read as "the limit as x approaches infinity of 1 over x is 0."
Infinity isn't a number
You can't plug ∞ into a formula: infinity isn't a real number. The arrow x → ∞ is shorthand for "x keeps growing without end." A limit at infinity describes where the outputs are heading on that endless trip, not a value at some last point.
Worked example: (2x + 1)/x
Split the fraction into two pieces (fine for any x other than 0):
(2x + 1)/x = 2x/x + 1/x = 2 + 1/x
As x grows, 1/x fades to 0, so the whole expression heads to 2 + 0 = 2:
lim x→∞ (2x + 1)/x = 2
Check with a big number: at x = 1000, 2001/1000 = 2.001. Close to 2, just above it.
On a graph, the curve flattens out along the line y = 2. A line like that, which a graph levels off along as x heads to infinity, is called a horizontal asymptote.
Try it yourself
Use the same split-the-fraction trick on these (answers below each):
lim x→∞ 5/x— the top stays 5 while the bottom grows without end. (Answer: 0.)lim x→∞ (3x + 5)/x— split it into3 + 5/x. (Answer: 3. Check: at x = 1000, 3005/1000 = 3.005.)
Good to know
- "The limit as x approaches infinity is L" means the values of f(x) get as close as we like to L for all large enough x.
- Infinity is not a real number;
x → ∞means x grows without bound. - For positive x,
1/xstays positive, so it approaches 0 without ever reaching it. - The line
y = 2is a horizontal asymptote of(2x + 1)/x. - The AP Calculus Course and Exam Description covers limits at infinity in Unit 1, Limits and Continuity; OpenStax covers them in section 4.6.
Answer to the comment question
lim x→∞ (4x + 7)/(2x) = ? The answer is 2. Split it: (4x + 7)/(2x) = 4x/(2x) + 7/(2x) = 2 + 7/(2x). As x grows, 7/(2x) fades to 0, so the limit is 2. Check: at x = 1000, 4007/2000 = 2.0035.
Test your knowledge
Quick quiz: Limits at infinity
4 multiple-choice questions. Pick one answer for each, then check how you did.
Sources
- OpenStax, "Calculus Volume 1, 4.6 Limits at Infinity and Asymptotes": definition of a limit at infinity; lim x→∞ 1/x = 0; y = L is a horizontal asymptote
- College Board, "AP Calculus AB and BC Course and Exam Description": Unit 1 "Limits and Continuity" covers limits at infinity and horizontal asymptotes
- Wolfram MathWorld, "Limit": limits as x → ∞
- Britannica, "Limit": a limit describes the value a function approaches; infinity is not a number to plug in
Video transcript
What happens to a graph when x gets huge?
Take one over x. Plug in ten, you get one tenth. A hundred, one hundredth. A thousand, one thousandth.
The answers shrink toward zero but never reach it. So the limit as x approaches infinity of one over x is zero.
Infinity isn't a number you plug in. It just means x keeps growing without end.
Now try two x plus one, over x. Split it into two, plus one over x.
As x grows, one over x fades to zero, so the whole thing heads to two. The curve flattens out along the line y equals two.
Your turn. Four x plus seven, over two x. What's the limit as x approaches infinity? Comment your answer below!
That's your byte.