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L'Hôpital's Rule

L'Hôpital's Rule
L'Hospital's Rule

A technique used to evaluate limits of fractions that evaluate to the indeterminate expressions The fraction 0/0, with 0 in the numerator and 0 in the denominator. and ±∞ divided by ±∞, an indeterminate expression showing infinity over infinity with sign variations.. This is done by finding the limit of the derivatives of the numerator and denominator.

Note: Most limits involving other indeterminate expressions can be manipulated into fraction form so that l'Hôpital's rule can be used.

 

L'Hôpital's Rule:

If f and g are differentiable on an open interval containing a such that g(x) ≠ 0 for all x ≠ a in the interval, and if either

The limit of f(x) as x approaches a equals 0 and The limit of g(x) as x approaches a equals 0

or

The limit of f(x) as x approaches a equals plus or minus infinity and The limit of g(x) as x approaches a equals plus or minus infinity

then

L'Hôpital's Rule formula: limit as x→a of f(x)/g(x) equals limit as x→a of f'(x)/g'(x)

Example:

 Example: lim(x→0) (1−cos x)/x² = lim(x→0) sin x/2x = lim(x→0) cos x/2 = 1/2