 ## Thursday, April 4, 2013

### Indeterminate Form - Combined

Category: Differential Calculus, Algebra

"Published in Suisun City, California, USA"

Evaluate

Solution:

Consider the given equation  above

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is ∞/∞, then we can use the L'Hopital's Rule but the problem is there's an exponent that associated with the Indeterminate Form. We have to eliminate the exponent first before we can apply the L'Hopital's Rule to the given equation. We cannot accept as a final answer for the given equation.

Let

Take natural logarithm on both sides of the equation

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is ∞/∞, then we can use the L'Hopital's Rule but the problem is the Indeterminate Form is a function of natural logarithm. We have to rewrite the above equation as follows

Divide both the numerator and denominator by 1/x, we have

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is ∞∙0, then we have to rewrite the above equation as follows

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is 0/0, then we can use the L'Hopital's Rule as follows

Substitute the value of x to the above equation, we have

Take the inverse natural logarithm on both sides of the equation

Therefore,