## Indeterminate Forms

## Indeterminate Form Type 0/0 & + ∞/∞

One instance when L’Hospital’s Rule is used when the limit as x approaches a of f(x) over g(x) where both f(x) and g(x) approach 0. Thus the limit may or may not exist and we have 0/0. This 0/0 is called the indeterminate form of type 0/0. Another instance L’Hospital’s Rule is used when the indeterminate form of type

__+__∞/∞. This occurs when the limit as x approaches a of f(x) over g(x) where both f(x) and g(x) approach__+__∞.## Indeterminate Form Type Products 0 *∞

The limit of product fg when the limit as x approaches a of f(x) times g(x) where f(x) and g(x) is 0 or ∞, is called the indeterminate form of type 0 * ∞. In this case we must rewrite the product fg as a quotient by placing either f(x) or g(x) in the denominator. This results back into the indeterminate form of type 0/0 or

__+__∞/∞ where L’Hospital’s Rule can be applied. The simpler function is preferred to be in the denominator.## Indeterminate Form Type Differences [f(x)- g(x)]

If the limit as x approaches a of f(x) –g(x) where f(x)=g(x)=∞, then the limit is called the indeterminate form of type ∞-∞. This again is a dilemma deciding which infinity is larger. Once again we must turn the difference into a quotient by using common denominator, rationalization, or factoring out a common factor. By multiplying by a common denominator, the two separate functions can be combined into one single quotient which results in indeterminate form 0/0 or

__+__∞/∞. From there L’Hospital’s Rule can be applied.## Indeterminate Form Type Powers

Many indeterminate forms can arise from the limit brought to a power.

For cases like these, we can either take the natural logarithm or rewrite the function as an exponential. For instance once the function is written in ln form, we must convert it into the indeterminate product, from the indeterminate product, we change it into indeterminate form 0/0 or

__+__∞/∞. From there we have found the limit as x approaches of lny, the last step is to solve for y by exponentiation.