L’Hôpital’s Rule

By direct substitution you obtain the indeterminate form of type 0/0. L’Hopital’s Rule Analytically evaluate the following limit: By direct substitution you obtain the indeterminate form of type 0/0. 0/0, ∞/∞, ∞-∞, 1^∞, ∞^0 and 0*∞ are called indeterminate forms because they do not indicate what the limit is or whether a limit exists

L’Hôpital’s Rule also works for right and left handed limits. 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 Then: As long as is finite or infinite. L’Hôpital’s Rule also works for right and left handed limits.

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

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

Example 2 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 applies since this is an indeterminate form. Differentiate the numerator and the denominator. Since the result is finite or infinite, the result is valid. Find the limit of the quotient of the derivatives.

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

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

L’Hôpital’s Rule does NOT apply since 0 is NOT an indeterminate form. 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.