L'Hôpital's Rule - Evaluating Indeterminate Forms

Calculus 1 · Axiom Academy

When a limit collapses to 0/0 or , differentiating the top and bottom turns an indeterminate ratio into one you can read off. Both and g(x)=x run into the origin, so at x=0 the quotient is 0/0 — undefined as written. But near x=0 each function hugs its tangent line , and the ratio of those two slopes is something definite. That ratio is the limit. 2. Substitute First — Only 0/0 or L'Hôpital's Rule is licensed by one thing: an indeterminate form . Before you differentiate anything, plug in x=a . If you get a determinate value like 1/0 , the rule simply does not apply — and using it anyway gives a wrong answer. 3. If It's Still Indeterminate, Apply It Again One pass of the rule doesn't always finish the job. Take : it's 0/0 , so differentiate to get — which is still 0/0 at x=0 . The fix is to recheck the form and, if it's indeterminate again, differentiate again. 4. Reaching Other Indeterminate Forms The rule only sees fractions, so other indeterminate forms — , , 0^0 , , — must first be rewritten as a quotient that lands on 0/0 or . L'Hôpital's Rule reads an indeterminate ratio as a race between two derivatives — but only when substitution actually gives 0/0 or . Scroll up to revisit any step.

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