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Evaluate
Solution:
To get the value of a given function, let's substitute the value of x 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 a term with the highest degree which is x2 and simplify the given equation as follows
Substitute the value of x 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 x as follows
Substitute the value of x to the above equation, we have
Again, apply the L'Hopital's Rule to the above equation, we have
Did you notice that the final equation is similar to the original equation? If you will continue this process, there will be an endless repetition of the process. Instead, let's consider the equation after the first application of L'Hopital's Rule as follows
Let's rewrite the right side of the equation by including the numerator into the radical at the denominator as follows
Perform the division of polynomials inside the radical, we have
Substitute the value of x to the above equation, we have
Therefore,
