L'Hopital's Rule Quiz

Grade 12 · Math · Quiz

10 questions on identifying indeterminate forms and applying L'Hopital's Rule.

Questions

1. What indeterminate form does lim(x→1) (x² - 1)/(x - 1) produce before applying L'Hopital?

   1. 1/0
   2. 0/0
   3. ∞/∞
   4. 0·∞
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   Answer: 0/0

   Substituting x=1: (1-1)/(1-1) = 0/0.

2. Evaluate lim(x→0) tan(x)/x.

   Show answer

   0/0 form. L'Hopital: lim sec²(x)/1 = sec²(0) = 1.

3. Evaluate lim(x→∞) ln(x)/x.

   1. 0
   2. 1
   3. ∞
   4. e
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   Answer: 0

   ∞/∞ form. L'Hopital: lim (1/x)/1 = 0.

4. L'Hopital's Rule requires differentiating f/g using the quotient rule.

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   You differentiate f and g separately (f′ and g′), then form f′/g′. Do NOT use the quotient rule.

5. Evaluate lim(x→0) (sin 3x)/(sin 2x).

   1. 1
   2. 3/2
   3. 2/3
   4. 0
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   Answer: 3/2

   0/0 form. L'Hopital: lim 3cos(3x)/(2cos(2x)) = 3(1)/(2(1)) = 3/2.

6. Evaluate lim(x→∞) x²/eˣ.

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   Apply L'Hopital twice: lim x²/eˣ → lim 2x/eˣ → lim 2/eˣ = 0.

7. Which form does lim(x→0⁺) x·ln(x) produce?

   1. 0/0
   2. ∞/∞
   3. 0·∞
   4. 1^∞
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   Answer: 0·∞

   x→0⁺ gives x→0 and ln(x)→-∞, so the product is 0·(-∞), a 0·∞ form.

8. Evaluate lim(x→0⁺) x·ln(x).

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   Rewrite as ln(x)/(1/x) — ∞/∞ form. L'Hopital: (1/x)/(-1/x²) = -x → 0.

9. If L'Hopital's Rule is applied and the new limit is still indeterminate, you can apply it again.

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   L'Hopital's Rule can be applied repeatedly as long as the form remains 0/0 or ∞/∞.

10. Evaluate lim(x→∞) (3x² + 1)/(x² - 2) using L'Hopital's Rule.

    1. 0
    2. 3
    3. 1
    4. ∞
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    Answer: 3

    ∞/∞ form. L'Hopital: lim 6x/(2x) = 3. (Or by dominant terms: 3x²/x² = 3.)

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