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L'Hôpital's Rule Calculator

L'Hôpital's rule demo. Enter f, g and a; for a 0/0 form compare lim f/g with lim f′/g′ to verify the rule—for indeterminate limits.

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0/0 form: lim f/g = lim f′/g′
Tends to 0 at a
Tends to 0 at a
The 0/0 point

📖 Tutorial｜L'Hôpital's Rule

1. Definition
For 0/0 or ∞/∞ indeterminate forms, lim f/g = lim f′/g′ (when the denominator derivative is nonzero).
2. Symbols
SymbolMeaning
f(x), g(x)Numerator and denominator
0/0Indeterminate type
f′/g′After L’Hôpital
aApproach point
3. How it works
  • We first verify the 0/0 form;
  • Compare f/g and f′/g′ numerically;
  • Agreement confirms the rule.
4. Steps
  1. Enter f and g;
  2. Enter the 0/0 point a;
  3. Click Calculate.
5. Example
Example: lim(x→0) sinx/x. Both →0; f′=cos→1, g′=1, so the limit is 1.
6. Pitfalls
Use only on 0/0 or ∞/∞ forms;
Re-check the indeterminate form each time;
L’Hôpital does not solve every limit.

❓ FAQ｜L'Hôpital's Rule

Which forms does it cover?
0/0 and ∞/∞. Other types (0·∞, ∞−∞) must first be rewritten into one of these.
What to watch out for?
Re-verify the indeterminate form each use and require g′≠0.
sinx/x as x→0 via L’Hôpital?
Numerator cos x→1, denominator 1, so the limit is 1.
Is it universal?
No. If f′/g′ diverges, the original limit may still exist; equivalent infinitesimals are sometimes faster.
Does it work for ∞/∞?
Yes; it covers both 0/0 and ∞/∞.