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Calculus IApplications

L’Hospital’s rule

Recognize an indeterminate form, choose a useful rewrite, and find the limit one step at a time. Then solve every example and practice question from the notes.

Indeterminate quotients · Step 1 of 9

The form alone cannot tell you the answer

x→0+{\displaystyle x\to0^+}
Same form, three different limits
QuotientAt this xLimit
x2x{\frac{x^2}{x}}0.5000{0}
3xx{\frac{3x}{x}}3.0003{3}
xx2{\frac{x}{x^2}}2.000∞{\infty}
An indeterminate form describes competing trends, not a number you can calculate. Two quantities both tending to zero can have a ratio tending to zero, a nonzero number, infinity, or no limit. Two quantities both growing without bound have the same problem.
Try it: Compare three quotients with the same form. Move closer to the target and watch their values separate.
Form
Approach the targetx = 0.5

The examples use positive x. A two-sided limit can fail too: ∣x∣x{\frac{|x|}{x}} tends to −1 on the left and 1 on the right. Never write “zero over zero” as a final answer.

Check yourself
Both numerator and denominator approach zero. What is the limit?

Need help choosing your next step in a limit?

Bring your practice questions. We work through them together until every type feels routine.

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