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Indeterminate Form, Infinity Over Infinity, 2

__Category__: Differential Calculus, Algebra

"Published in Vacaville, California, USA"

Evaluate
__Solution__:
To get the value of a given function, let's substitute the value of n to the above equation, we have

Since
the answer is -∞/∞, then it is an Indeterminate Form which is not
accepted as a final answer in Mathematics. We have to do something first
in the given equation so that the final answer will be a real number,
rational, or irrational number.
__Method 1__:
Since the answer is Indeterminate Form, then we have to divide both sides of the fraction by 2^{n} and simplify the given equation as follows
Substitute the value of n to the above equation, we have
Therefore,
__Method 2__:
Another
method of solving Indeterminate Form is by using L'Hopital's Rule. This
is the better method especially if the rational functions cannot be
factored. L'Hopitals Rule is applicable if the Indeterminate Form is
either 0/0 or ∞/∞. Let's apply the L'Hopital's Rule to the given
function by taking the derivative of numerator and denominator with
respect to n as follows
Again, apply the L'Hopital's Rule to the above equation, we have
Since the resulting equation is the same as the given equation, then we cannot use the L'Hopital's Rule because of the repetitive solutions and results. There's no end in this process. In this case, we have to consider the Method 1 in solving the given limits.