Limits at infinity
As x grows
What happens to \dfrac{3x^2+2x-1}{x^2+5} when x is a million? The numerator and the denominator both become enormous. Does their ratio become enormous too, or does it settle down?
Zoom out on the graph and watch which terms still matter. The indigo dashes show the ratio of the leading terms alone. How far out do you need to go before the two curves are hard to tell apart?
The window runs from −R to R. Drag the slider to zoom out.
| x | f(x) | f(−x) |
|---|
Divide by the highest power
Every term of the form \tfrac{c}{x^p} with p>0 approaches 0 as x\to\pm\infty. So divide the numerator and the denominator by the highest power of x in the denominator, and most terms disappear:
The line y=3 is a horizontal asymptote. Doing this for any rational function gives three cases, decided by the degrees of the top and bottom.
| Degrees | Limit as x → ±∞ | Horizontal asymptote |
|---|---|---|
| top < bottom | 0 | y = 0 |
| top = bottom | ratio of the leading coefficients | y = that ratio |
| top > bottom | grows without bound (±∞) | none |
Why should only the leading terms matter? At x=1000, the term 3x^2 is three million, while 2x-1 is under two thousand.
Roots need care. Choose \tfrac{\sqrt{4x^2+1}}{x+1} in the graph and compare f(x) with f(-x). Because \sqrt{x^2}=|x|, the limits at \infty and -\infty are different.
Solve it step by step
Find each limit at infinity. Type inf or -inf when the function grows without bound. The score counts steps you answer correctly on the first try.
Common mistakes
- Writing ∞/∞ = 1
Like \tfrac00, the form \tfrac{\infty}{\infty} is indeterminate. It tells you to simplify, not what the answer is.
- Forgetting the sign at −∞
An odd power of a negative number is negative. When the top's degree is higher, check the sign of the leading terms for x\to-\infty separately.
- Treating √(x²) as x
For negative x, \sqrt{x^2}=-x. This is where different limits at the two ends usually come from.