Unit 5: Calculus
Topic 5.13: Limits & l'Hopital (HL only) Questions
Practice 20 exam-style questions for IB Math AA SL Topic 5.13. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.
2Evaluate3 marks
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Use l'Hopital's rule to evaluate the limit of ( − 1)/x as x → 0.
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L'Hopital's rule may be used only when the limit is of the form:
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lim as x→1 of (x² − 1)/(x − 1), by L'Hopital, is:
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A limit of the form 0·∞ should first be rewritten as:
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You stop applying L'Hopital's rule when:
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Use l'Hopital's rule to evaluate the limit of (ln x)/(x − 1) as x → 1.
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Evaluate the limit of (sin 5x)/(sin 2x) as x → 0 using l'Hopital's rule.
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Applying L'Hopital to lim (sin x)/x means differentiating to get:
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Use l'Hopital's rule to evaluate the limit of (arctan x)/(3x) as x → 0.
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If lim (x→0) of (top)/x³ is finite, then as x→0 the top must:
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The limit of (sin²(kx))/x² as x → 0 is 9. Find the possible values of k.
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Use l'Hopital's rule to evaluate the limit of (eˣ − x − 1)/x² as x → 0.
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Evaluate the limit of (cos x − 1)/x² as x → 0.
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Evaluate the limit of (1 − cos x)/x² as x → 0, applying l'Hopital where appropriate.
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2026
The most common mistake when using L'Hopital's rule is:
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2026
Find the limit of x·cot x as x → 0 (rewrite as a fraction first).
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