Math Help

Indeterminate Forms - Types, Procedure to Evaluate Them

In a Nut Shell: Sometimes the evaluation of the limit of a function that involves quotients,

differences, products, or powers requires special attention. The value of the limit is

indeterminate and might take on any value. You cannot simply put in the value of x

in the expression for f(x) and g(x) and come up with the value of the limit.

There are four types that arise. The procedure to establish the limit depends on the type.

Each type appears below with an example illustrating the recommended procedure.

Type 1: (quotient) The limit of the function involves 0/0 or ∞/∞

In this type you can use L’Hospital’s Rule where you take the derivative of the

numerator and the derivative of the denominator separately and recalculate the

limit. In some problems you may need to repeat L’Hospital’s rule more than once

since the limit may still be indeterminate following each application of the rule.

Procedure: lim ( f(x) )/(g(x) ) = lim ( f ’(x) )/(g ’(x) )

x → 0 x → 0

Example: lim ( sin x )/x = lim ( cos x )/ 1 = 1

x → 0 x → 0

Type 2: (difference) The limit of the function involves ∞ - ∞

Procedure: Convert to a quotient 0/0 or ∞/∞ by algebraic manipulation.

This is perhaps the most difficult type since there is no standard procedure to

accomplish the manipulation.

Example: lim ( 1/x - 1/sinx ) = lim [(sin x – x)/ (x sin x)]

x → 0 x → 0

where the difference is multiplied and divided by x sin x resulting in the

indeterminate for 0 / 0 . So use L’Hospital’s Rule.

lim [(cos x – 1)/ (sin x + x cos x)] Use L’Hospital’s rule again.

x → 0

lim [(- sin x )/ (2 cos x - x sin x)] = 0

x → 0