LIMIT OF A FUNCTION

Many of us students dislikes mathematics because its complex nature in terms of operation. That is why this page was made to depict what was possible to make it vivid.

We were provided five problems to solve. It is from easy to a bit harder. The operation shows gradual development to show what is the real deal. It is recommended that the student has basic idea about algebraic operation in this case, but if the idea is intermediate or higher, the better. Below were the samples that usually found in any mathematical books and in the classrooms. Note that the given functions were highlighted red, and green for answers. It is by the authors discretion to do such to make the readers brain plots the necessary requirements to arrive in the right answer. If typographical error occurs, please let us know in the comment box.


Illustrations:

Perform the indicated operations with the given functions.

$IF,{\color{DarkRed} f(a)=\frac{a-a^{2}}{1+a^{2}}},find;f(0),f(-2),f(-\frac{1}{2}),f(-2b),f(tan(y))$

$1.){\color{DarkRed} f(a)=\frac{a-a^{2}}{1+a^{2}}},findf(0)$

   $f(0)=\frac{0-0^{2}}{1+0^{2}}$

   $f(0)=\frac{0}{1}=0,{\color{DarkGreen} ans}$

$2.){\color{DarkRed} f(a)=\frac{a-a^{2}}{1+a^{2}}},find;f(-2)$

$f(-2)=\frac{2-(-2)^{2}}{1+(-2)^{2}}$

$f(-2)=\frac{-2-4}{1+4}=\frac{-6}{5},{\color{DarkGreen} ans}$

   $3.){\color{DarkRed} f(a)=\frac{a-a^{2}}{1+a^{2}}},find;f(\frac{1}{2})$

$f(\frac{1}{2})=\frac{\frac{1}{2}-(-\frac{1}{2})^{2}}{1+(-\frac{1}{2})2}$

$f(\frac{1}{2})=\frac{\frac{1}{2}-\frac{1}{4}}{1+\frac{1}{4}}$

$f(\frac{1}{2})=\frac{\frac{1}{4}}{\frac{5}{4}}$

$f(\frac{1}{2})=\frac{1}{4}\ast \frac{4}{5}=\frac{1}{5}, {\color{DarkGreen} ans}$

   $4.){\color{DarkRed} f(a)=\frac{a-a^{2}}{1+a^{2}}},find;f(-2b)$

   $f(-2b)=\frac{2b-(-2b)^{2}}{1+(-2b)^{2}}$

   $f(-2b)=\frac{2b-4b^{2}}{1+4b^{2}}$

   $f(-2b)=\frac{2b(1-2b)}{1+4b^{2}},{\color{DarkGreen} ans}$

   $5.){\color{DarkRed} f(a)=\frac{a-a^{2}}{1+a^{2}}},find;f(tan(y))$

$f(tany)=\frac{tan(y)-tan^{2}(y)}{1+tan(y)}$

$f(tany)=\frac{tan(y)(1-tan^{2}(y))}{1+tan^{2}(y)}$

$f(tany)=\frac{tany(1-tan^{2}y)}{sec^{2}y}$

$f(tany)=\frac{\frac{siny}{cosy}(1-\frac{siny}{cosy})}{\frac{1}{cos^{2}y}}$

$f(tany)=\frac{\frac{siny}{cosy}(\frac{cosy-siny}{cosy})}{\frac{1}{cos^{2}y}}$

$f(tany)=siny(cosy-siny),{\color{DarkGreen} ans}$

More Examples:

$Given; g(x)= 3x^{5}-4x^{4}+2x^{2}-x+1, Solve for; f({\color{Red} -2}),f({\color{Red} -2a}),f({\color{Red} 3ab})$