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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 4

Lesson 7: Using L’Hôpital’s rule for finding limits of indeterminate forms

# Proof of special case of l'Hôpital's rule

L'Hôpital's rule helps us find limits in the form limit, start subscript, x, \to, c, end subscript, start fraction, u, left parenthesis, x, right parenthesis, divided by, v, left parenthesis, x, right parenthesis, end fraction where direct substitution ends in the indeterminate forms start fraction, 0, divided by, 0, end fraction or start fraction, infinity, divided by, infinity, end fraction.
The rule essentially says that if the limit limit, start subscript, x, \to, c, end subscript, start fraction, u, prime, left parenthesis, x, right parenthesis, divided by, v, prime, left parenthesis, x, right parenthesis, end fraction exists, then the two limits are equal:
limit, start subscript, x, \to, c, end subscript, start fraction, u, left parenthesis, x, right parenthesis, divided by, v, left parenthesis, x, right parenthesis, end fraction, equals, limit, start subscript, x, \to, c, end subscript, start fraction, u, prime, left parenthesis, x, right parenthesis, divided by, v, prime, left parenthesis, x, right parenthesis, end fraction
The AP Calculus course doesn't require knowing the proof of this fact, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.