__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 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 x^{2} 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,

__Category__: Differential Calculus, Trigonometry

"Published in Vacaville, California, USA"
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 0/0, 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 apply the double angle formula at the numerator and then simplify 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
Therefore,
**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 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 cannot use the L'Hopital's Rule because the Indeterminate form is (∞ - ∞)/∞.
L'Hopital's Rule is applicable if the Indeterminate Form is either 0/0
or ∞/∞. We have to do something first in the given equation so that the
Indeterminate Form becomes 0/0 or ∞/∞.
Since the third degree polynomials in the numerator and denominator have no factors or cannot be factored, then we have to divide both sides of the fraction by the highest degree variable which is x^{3} as follows
Substitute the value of x to the above equation, we have
Therefore,
__Category__: Differential Calculus, Algebra

"Published in Newark, California, USA"
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 0/0, 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 factor the numerator and denominator if they can and then simplify 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
Therefore,