Understanding Limits: The Foundation of Calculus

A limit describes the value a function approaches, not necessarily the value it takes.

$$\underset{x\to 1}{\lim }\frac{x^2-1}{x-1}$$

At $x=1$ the expression is $\frac{0}{0}$ — undefined. But for every $x\ne 1$ it equals $x+1$, which approaches $2$. So the limit is $2$.

A strategy that works

1. Substitute directly. If you get a number, you're done.
2. 0/0? Factor and cancel, rationalize, or use a trig identity.
3. Still stuck? Apply L'Hôpital's rule: differentiate numerator and denominator separately.
4. At infinity? Compare the highest powers.

When limits don't exist

If the left- and right-hand limits differ, as with ${\lim }_{x\to 0}\frac{1}{x}$, the two-sided limit does not exist.

Try the Limit Calculator.