The formula
if substituting gives 0/0 or ∞/∞, differentiate top and bottom
Why it works
A limit is not the value of the function at the point: it is what the function approaches as x approaches. That is why it is worked out from both sides, and why it can exist even when the function is undefined there — the classic case is sin(x)/x at zero, which is one even though the expression reads 0/0. If the two sides disagree the limit does not exist, and saying so matters as much as giving a number. When a quotient gives 0/0 or ∞/∞, L’Hôpital’s rule lets you differentiate numerator and denominator separately as many times as needed.
How to solve it by hand
- Substitute the point for x and see what comes out
- If it gives a number, the function is continuous there and that is the limit
- If it gives 0/0 or ∞/∞, differentiate top and bottom and substitute again
- Check both sides: if they disagree, the limit does not exist
What is worth knowing
L’Hôpital’s rule only applies to the indeterminate forms 0/0 and ∞/∞: using it on 0·∞ or ∞−∞ without turning them into a quotient first is the commonest mistake, and it gives false answers. When there is no quotient indeterminacy, this page approaches the point numerically and says the figure is approximate: that is the difference between an exact result and a good estimate, and it is worth knowing which one you are looking at. A function can have an infinite limit from both sides — 1/x² at zero — and then the limit exists and is infinity; if the signs differ, as in 1/x, it does not.