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Evaluate
Solution:
Consider the given equation
Substitute the value of x to the above equation, we have
Since the answer is ∞0, then it is also another type of Indeterminate Form and it is not accepted as a final answer in Mathematics. We know that any number raised to zero power is always equal to one except for infinity that's why it is also an Indeterminate Form. In this type of Indeterminate Form, we cannot use the L'Hopital's Rule because the L'Hopital's Rule is applicable for the Indeterminate Forms like 0/0 and ∞/∞. Since the given equation is exponential equation, let's consider the following procedure
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 0∙∞, 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 ∞/∞, then we can now use the L'Hopital's Rule as follows
Substitute the value of x to the above equation, we have
Since the Indeterminate Form is again ∞/∞, then we have to use the L'Hopital's Rule again as follows
Substitute the value of x to the above equation, we have
Since the Indeterminate Form is again ∞/∞, then we have to use the L'Hopital's Rule again as follows
Take inverse natural logarithm on both sides of the equation
Therefore,
