Calculus I · Unit 2B · review

L'Hopital's Rule Review

Review

Summary

Use L'Hopital only for $0/0$ or $\infty/\infty$ quotients after verifying hypotheses. Recheck the form after each application. Transform products, differences, and powers before using the rule.

Exercise

Evaluate $\lim_{x\to0}(e^{2x}-1)/x$ .

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Exercise 1 answer

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Exercise

Evaluate $\lim_{x\to0}(\sin x-x)/x^3$ .

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Exercise 2 answer

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Exercise

Evaluate $\lim_{x\to\infty}x^5/e^x$ .

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Exercise 3 answer

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Exercise

Evaluate $\lim_{x\to\infty}\ln x/\sqrt{x}$ .

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Exercise 4 answer

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Exercise

Evaluate $\lim_{x\to0^+}x^2\ln x$ .

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Exercise 5 answer

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Exercise

Evaluate $\lim_{x\to\infty}(\sqrt{x^2+x}-x)$ and compare rationalization with L'Hopital.

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Exercise 6 answer

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Exercise

Evaluate $\lim_{x\to0^+}x^x$ .

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Exercise 7 answer

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Exercise

Explain why L'Hopital is invalid for $\lim_{x\to0}(2+x)/(3-x)$ .

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Exercise 8 answer

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After the explanation

Use the section idea

Reading lens

Identify the limiting form before differentiating; transformation and simpler limit laws come before L'Hopital's Rule.

Mental model

The rule compares numerator and denominator growth only for verified zero-over-zero or infinity-over-infinity quotients.

Decision

Evaluate numerator and denominator limits separately, transform nonquotient forms, apply the rule only when justified, then recheck.

Common trap

Using L'Hopital because an expression looks difficult rather than because the required indeterminate quotient has been proved.

Check yourself

Can you name the form at every application and explain why direct substitution or algebra is not already enough?

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