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Difficulty guide

The label describes the problem, not your ability.

1/5 - Foundational

Warm-up setup; checks one basic idea.

2/5 - Routine exam skill

Normal syllabus skill; usually one main rule or interpretation.

3/5 - Standard multi-step

Exam-style problem with multiple steps or a common trap.

4/5 - Hard exam skill

High-end exam problem; combines methods carefully.

5/5 - Challenge

Stretch problem beyond normal exam pressure.

Question

Evaluate the following limits. Justify each use of L'Hospital's Rule by identifying the indeterminate form.
Total: 6 points

(a) lim⁡x→0ex−1−xx2\displaystyle \lim_{x\to0}\frac{e^x-1-x}{x^2} (2 points)

(b) lim⁡x→∞ln⁡xx\displaystyle \lim_{x\to\infty}\frac{\ln x}{x} (2 points)

(c) Explain why L'Hospital's Rule is not directly applicable to lim⁡x→0x+1x\displaystyle \lim_{x\to0}\frac{x+1}{x}. (2 points)

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