__Category__: Differential Calculus, Algebra"Published in Newark, California, USA"

Evaluate the limit for

__Solution__:

Consider the given equation, substitute the value of x, we have

Since the answer is ∞ - ∞ which is also another type of Indeterminate Form, it is not accepted in Mathematics as a final answer. In this type of Indeterminate Form, you cannot use the L'Hopital's Rule because the L'Hopital's Rule is applicable for the Indeterminate Forms like 0/0 and ∞/∞. In this case, let's rewrite the given equation as follows

Substitute the value of x to the above equation

Since the Indeterminate Form at this time is 0/0, we can now apply the L'Hopital's Rule

As you notice that when you substitute the value of x to the above equation, the answer will be 0/0 again. If you will apply the L'Hopital's Rule to the above equation, the equation will be more complicated and the answer will be 0/0 again. In this case, we have to rewrite the above equation as follows

We cannot apply the L'Hopital's Rule for the above equation since the numerator will be ∞ - ∞ when x equals ∞. If you will rewrite the above equation, the trend of rewriting the equation will be the same over and over and it will be more complicated as well.

But don't worry, we have a special solution for the exponential and logarithmic functions whose Indeterminate Forms are ∞ - ∞. Let's consider this one

Let

so that

If x equals ∞, the above equation will be

Since the Indeterminate Form is ∞/∞, then we can apply the L'Hopital's Rule for the above equation as follows

Since

then it follows that

Therefore,