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Quiz: L'Hôpital's Rule and Applications

Test your understanding of L'Hôpital's Rule with these review questions.

1. L'Hôpital's Rule applies to which indeterminate forms?

  1. 0/0 only
  2. ∞/∞ only
  3. 0/0 and ∞/∞
  4. All indeterminate forms directly
Show Answer

The correct answer is C. L'Hôpital's Rule applies directly to 0/0 and ∞/∞ forms. Other indeterminate forms (0·∞, ∞−∞, etc.) must first be rewritten in one of these forms.

Concept Tested: L'Hôpital's Conditions

2. L'Hôpital's Rule states that if lim f(x)/g(x) is indeterminate, then it equals:

  1. lim [f(x) − g(x)]
  2. lim [f'(x)/g'(x)]
  3. lim [f(x) · g(x)]
  4. f'(a)/g'(a)
Show Answer

The correct answer is B. L'Hôpital's Rule: If lim f(x)/g(x) gives 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists.

Concept Tested: L'Hôpital's Rule

3. What is lim(x→0) sin(x)/x using L'Hôpital's Rule?

  1. 0
  2. 1
  3. ∞
  4. Does not exist
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The correct answer is B. This is 0/0. Apply L'Hôpital's: lim(x→0) cos(x)/1 = cos(0)/1 = 1. This confirms the fundamental limit we proved earlier with the Squeeze Theorem.

Concept Tested: Zero Over Zero Apply

4. What is lim(x→∞) x²/eˣ?

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

The correct answer is B. This is ∞/∞. Apply L'Hôpital's twice: lim(x→∞) 2x/eˣ (still ∞/∞), then lim(x→∞) 2/eˣ = 0. Exponentials always beat polynomials as x→∞.

Concept Tested: Infinity Over Infinity Apply

5. To use L'Hôpital's Rule on lim(x→0) x · ln(x), you should first:

  1. Apply L'Hôpital's directly
  2. Rewrite as ln(x)/(1/x) to get ∞/∞ form
  3. Evaluate by substitution
  4. Factor out x
Show Answer

The correct answer is B. The form 0 · (−∞) is indeterminate but L'Hôpital's doesn't apply directly. Rewrite as ln(x)/(1/x) = ln(x)/(x⁻¹) to get −∞/∞ form, then apply L'Hôpital's.

Concept Tested: Zero Times Infinity

6. What is lim(x→∞) (1 + 1/x)ˣ?

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

The correct answer is B. This is the 1^∞ form. Take ln: ln(y) = x·ln(1 + 1/x). Rewrite and apply L'Hôpital's to get limit 1. So y = e¹ = e.

Concept Tested: One to Infinity

7. When applying L'Hôpital's Rule, you differentiate:

  1. The entire fraction using the Quotient Rule
  2. The numerator and denominator separately
  3. Only the numerator
  4. Only the denominator
Show Answer

The correct answer is B. You differentiate the numerator and denominator separately, not as a quotient. The rule is lim f/g = lim f'/g', not lim [d/dx(f/g)].

Concept Tested: L'Hôpital's Rule

8. What is lim(x→0) (eˣ − 1)/x?

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

The correct answer is B. This is 0/0 form. Apply L'Hôpital's: lim(x→0) eˣ/1 = e⁰ = 1. This limit is important as the definition of the derivative of eˣ at x = 0.

Concept Tested: Zero Over Zero Apply

9. If applying L'Hôpital's Rule once still gives an indeterminate form, you should:

  1. Conclude the limit doesn't exist
  2. Try a different method
  3. Apply L'Hôpital's Rule again
  4. Substitute x = 0
Show Answer

The correct answer is C. You can apply L'Hôpital's Rule repeatedly as long as the result is still 0/0 or ∞/∞ and the conditions are satisfied. Eventually you should reach a determinate form.

Concept Tested: Repeated L'Hôpital

10. What must you verify before applying L'Hôpital's Rule?

  1. That f(a) and g(a) both equal zero
  2. That the limit is of the form 0/0 or ∞/∞
  3. That f and g are polynomials
  4. That the answer is finite
Show Answer

The correct answer is B. Before applying L'Hôpital's Rule, verify that direct substitution gives an indeterminate form (0/0 or ∞/∞). Applying the rule to non-indeterminate forms gives wrong answers.

Concept Tested: Verify L'Hôpital