Limit Rules
A copy-able sheet — limit laws, the standard limits, L’Hôpital’s rule, continuity, and asymptotes.
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Unless a limit point is written explicitly, every law above holds as x → a for any a where both limits on the right exist. Try direct substitution first: the laws only earn their keep once substitution gives an indeterminate form.
Start every limit with direct substitution — if the function is continuous there, the limit is just f(a). The rules matter when substitution gives an indeterminate form such as 0/0. The one most often needed is lim(x→0) sin x ÷ x = 1, which substitution alone cannot give.
Substitution first, rules second
A limit asks what a function approaches, not what it equals, and for continuous functions those are the same thing — which is why plugging the value in works so often. The machinery on this page exists for the cases where it does not: a denominator that vanishes, a difference of two infinities, a function defined piecewise across the point in question.
When substitution fails, the route depends on what it produced. A 0/0 from a rational function usually means a shared factor waiting to be cancelled, or a conjugate waiting to be multiplied through. A limit at infinity usually means dividing numerator and denominator by the highest power of x. Only when algebra runs out does L’Hôpital’s rule earn its place.
Why one-sided limits matter
A two-sided limit exists only when both sides agree. For f(x) = |x| ÷ x the left limit is −1 and the right limit is +1, so the limit at 0 does not exist even though the function is perfectly well behaved on either side. Any function defined by cases needs both sides checked at the boundary before you can claim a limit at all.
The indeterminate forms
These are the only forms that need work. Anything else substitution already answered.
| Form | What to do |
|---|---|
| 0/0 | Factor and cancel, rationalise, or apply L’Hôpital’s rule |
| ∞/∞ | Divide through by the highest power, or apply L’Hôpital’s rule |
| 0 · ∞ | Rewrite as a quotient, then use L’Hôpital |
| ∞ − ∞ | Combine over a common denominator or rationalise |
| 1^∞ | Take the natural log, find that limit, then exponentiate |
| 0⁰ | Take the natural log, find that limit, then exponentiate |
| ∞⁰ | Take the natural log, find that limit, then exponentiate |
Horizontal asymptotes of a rational function
Compare the degree of the numerator with the degree of the denominator.
| Degrees | As x → ±∞ | Example |
|---|---|---|
| Numerator lower | y = 0 | (x + 1) ÷ (x² + 3) → 0 |
| Degrees equal | Ratio of leading coefficients | (3x² + 1) ÷ (2x² − x) → 3/2 |
| Numerator higher by 1 | No horizontal asymptote; a slant one instead | (x² + 1) ÷ x → x |
| Numerator higher by 2 or more | No horizontal or slant asymptote | (x³ + 1) ÷ x → ±∞ |
Where L’Hôpital’s rule goes wrong
Three errors are common. The first is applying it to a form that is not indeterminate: a limit of the shape 2/0 is not 0/0, and differentiating both parts changes the answer rather than revealing it. Check the form before differentiating, and check it again after each application.
The second is differentiating the quotient rather than the numerator and denominator separately — the rule is lim f′ ÷ g′, never the derivative of f ÷ g. The third is looping forever: for limits such as x ÷ √(x² + 1) the rule cycles back to where it started, and the algebraic route of dividing by x finishes in one step.