Explain and given an example of L'Hospital's Rule.  

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justaguide eNotes educator| Certified Educator

l'Hopital's (or l'Hospital's) Rule is used in determining limits for expressions that satisfy certain conditions.

For a limit of the form `lim_(x-> a) f(x)/g(x)` if substituting x = a gives `f(a)/g(a)` with an indeterminate form `0/0` , `oo/oo` , `1^oo` , `0^0` , `oo^0` , `0*oo` or `oo - oo` , the denominator and numerator can be substituted with their derivatives.

For example:

In determining the value of `lim_(x->0)(sin x)/x` , if x it substituted with 0 the result is the indeterminate form `0/0` . Here, using the l'Hopital's rule allows the denominator x and the numerator sin x to be substituted by their derivatives.

The result is `lim_(x->0) (cos x)/1` ; substituting x = 0, gives cos 0 = 1 which is the value of `lim_(x->0)(sin x)/x`.