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L’Hôpital’s Rule

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Another Look at Limits Analytically evaluate the following limit: This limit must exist because after direct substitution you obtain the indeterminate form of type 0/0. But this limit is to difficult to evaluate because the polynomials have degrees higher than 2.

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If… f and g are differentiable functions on an open interval containing x = a, except possibly at x = a. One of the following: and OR and Then: As long as is finite or infinite. L’Hôpital’s Rule L’Hôpital’s Rule also works for right and left handed limits.

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Procedure for L’Hôpital’s Rule To evaluate the limit: 1.Check that the limit of f(x)/g(x) as x approaches a is an indeterminate form of the type 0/0 or ∞/∞. 2.Differentiate f and g separately. 3.Find the limit of f '(x)/g '(x) as x approaches a. 4.If the limit is finite, +∞, or -∞, then it is equal to the limit in question.

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Example 1 Analytically evaluate the following limit: In order to use L’Hôpital’s Rule direct substitution must result in 0/0 or ∞/∞. Differentiate the numerator and the denominator. Find the limit of the quotient of the derivatives. L’Hôpital’s Rule applies since this is an indeterminate form. Since the result is finite or infinite, the result is valid.

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Example 2 Analytically evaluate the following limit: In order to use L’Hôpital’s Rule direct substitution must result in 0/0 or ∞/∞. Differentiate the numerator and the denominator. Find the limit of the quotient of the derivatives. L’Hôpital’s Rule applies since this is an indeterminate form. Since the result is finite or infinite, the result is valid.

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No Longer “Freebie” Limits The following limits were assumed to be true to assist in finding other limits: If you evaluate these limits with L’Hôpital’s Rule, you will no longer need to memorize them.

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Example 3 Analytically evaluate the following limit: In order to use L’Hôpital’s Rule direct substitution must result in 0/0 or ∞/∞. Differentiate the numerator and the denominator. Find the limit of the quotient of the derivatives. L’Hôpital’s Rule applies since this is an indeterminate form. This is still an indeterminate form, apply L’Hôpital’s Rule again to the new limit. Differentiate the new numerator and the denominator. Find the limit of the quotient of the second derivatives. Since the result is finite or infinite, the result is valid. NOTE: L’Hôpital’s Rule can be applied as many times as needed.

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Example 4 Analytically evaluate the following limit: In order to use L’Hôpital’s Rule direct substitution must result in 0/0 or ∞/∞. Differentiate the numerator and the denominator. Find the limit of the quotient of the derivatives. L’Hôpital’s Rule applies since this is an indeterminate form. Since the result is finite or infinite, the result is valid. Rewrite the expression as one ratio in order to use L’Hôpital’s Rule. Strategy: Rewrite the trig functions using cos x and sin 0.

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Example 5 Analytically evaluate the following limit: In order to use L’Hôpital’s Rule direct substitution must result in 0/0 or ∞/∞. L’Hôpital’s Rule does NOT apply since 0 is NOT an indeterminate form. This limit is actually easy to find because the function is continuous at π. We have already found the limit with direct substitution.

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