Question

If $f\left(x\right)=\frac{a\sin x-bx+c{x}^2+x^3}{2x^2\ell n(1+x)-2x^3+x^4}$, when $x\ne 0$ and $f\left(x\right)$ is continuous at $x=0$, find value of $200\times f\left(0\right)$

Answer 1

$\underset{x\to 0}{lim}\frac{a[x-\frac{x^3}{6}+\frac{x^5}{120}-.......]-bx+c{x}^2+x^3}{2x^2[x-\frac{x^2}{2}+\frac{x^3}{3}.........]-2x^3+x^4}$
$={lim}_{x\to 0}\frac{(a-b)x+c{x}^2+[1-\frac{a}{6}]x^3+\frac{a{x}^5}{120}+.....}{\frac{2x^5}{3}-\frac{x^6}{2}+......}$
for this limit to be exist, $a-b=0,c=0$, & $1-\frac{a}{6}=0$
$\Rightarrow a=b=6$ & $c=0$.
then $f\left(0\right)=\frac{6}{120}.\frac{3}{2}=\frac{3}{40}\Rightarrow 200f\left(0\right)=15$