Home

Limits at Infinity Calculator

Limits-at-infinity demo. Enter f(x) and direction to evaluate f at 10,100,1000... and read off the horizontal asymptote—for infinity limits.

展开更多 ▾
lim(x→+∞) f(x) = A ⇔ horizontal asymptote y = A
Supports + - * / ^ and sin cos tan sqrt exp log pi e
x to +∞ or −∞

📖 Tutorial｜Limits at Infinity

1. Definition
If f(x)→A as x→±∞, then lim(x→±∞)f(x)=A and y=A is a horizontal asymptote.
2. Symbols
SymbolMeaning
f(x)The function
+∞ / −∞Two infinite directions
AHeight of the horizontal asymptote
3. How it works
  • We evaluate f at ±10, ±100, ±1000, ±10000;
  • Watch whether the values settle to a constant A;
  • If so, A is the horizontal asymptote, else it diverges.
4. Steps
  1. Enter f(x);
  2. Choose x→+∞ or −∞;
  3. Click Calculate and read the last row as the asymptote.
5. Example
Example: f(x)=(3x+1)/x=3+1/x→3 as x→+∞, so the horizontal asymptote is y=3.
6. Pitfalls
For polynomials the leading term dominates at infinity;
For rational functions compare leading coefficients;
Oscillating functions like sinx have no limit at infinity.

❓ FAQ｜Limits at Infinity

What is a horizontal asymptote?
If lim(x→∞)f(x)=A, then y=A is the horizontal asymptote the curve approaches far away.
How do rational functions behave at infinity?
Compare the degrees of numerator and denominator: lower numerator →0, equal → leading-coefficient ratio, higher →∞.
Does sinx have a limit at infinity?
No. It oscillates between −1 and 1 and never settles.
Do +∞ and −∞ always give the same result?
No. e^x diverges at +∞ but tends to 0 at −∞; check both sides.
Why does my number blow up?
The function is genuinely growing (e.g. x²), so there is no horizontal asymptote.