L'Hopital's Rule: Indeterminate Forms (Calculus I)
Free printable Calculus I worksheet on L'Hopital's rule for 0/0 and infinity/infinity forms: factorable rational limits, sin, tan, exponential and logarithm limits at 0, and growth-rate comparisons at infinity. Every limit computed, with the indeterminate form verified in code.
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Calculus I · Math worksheet
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Math
L'Hopital's Rule: 0/0 and ∞/∞
First substitute to confirm the limit is a genuine indeterminate form (0/0 or ∞/∞); only then differentiate the numerator and denominator separately and take the limit again. Repeat if the new limit is still indeterminate.
- 1.Evaluate lim (x→-1) (3x² + 7x + 4) / (x² + 6x + 5). Verify the limit has the indeterminate form 00 before applying L'Hopital's rule.
- 2.Evaluate lim (x→0) (sin(4x)) / (6x). Verify the limit has the indeterminate form 00 before applying L'Hopital's rule.
- 3.Evaluate lim (x→∞) (8x² - 2x + 6) / (4x² + 5). Verify the limit has the indeterminate form ∞/∞ before applying L'Hopital's rule.
- 4.Evaluate lim (x→0) (sin(7x)) / (5x). Verify the limit has the indeterminate form 00 before applying L'Hopital's rule.
- 5.Evaluate lim (x→2) (2x² - 7x + 6) / (3x² - 7x + 2). Verify the limit has the indeterminate form 00 before applying L'Hopital's rule.
- 6.Evaluate lim (x→∞) (2x² - 5x - 6) / (6x² - 6x + 1). Verify the limit has the indeterminate form ∞/∞ before applying L'Hopital's rule.
- 7.Evaluate lim (x→0) (sin(4x)) / (3x). Verify the limit has the indeterminate form 00 before applying L'Hopital's rule.
- 8.Evaluate lim (x→∞) x² / . Verify the limit has the indeterminate form ∞/∞ before applying L'Hopital's rule.
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