Question

Solve $\underset{x\to 0}{lim}\frac{a\sin x-bx+c{x}^2+x^3}{2x^2\log (1+x)-2x^3+x^4}$, if it exists and is finite, also find $a,b,c$.

Answer 1

$\underset{x\to 0}{lim}\frac{asinx-bx+c{x}^2+x^3}{2x^{2}log(1+x)-2x^3+x^4}$

$=\underset{x\to 0}{lim}a(x-\frac{x^3}{6})-bx+c{x}^2+x^3$

$=\underset{x\to 0}{lim}\frac{asinx-bx+c{x}^2+x^3}{2x^{2}log(1+x)-2x^3+x^4}$

$=\underset{x\to 0}{lim}\frac{a(x-\frac{x^3}{3!}+\frac{x^5}{5!})-bx+c{x}^2+x^3}{2x^2.x-2x^3+x^4}$

$\underset{x\to 0}{lim}\frac{(a-b)x+c{x}^2+x^3(1-\frac{a}{6})+\frac{a{x}^5}{120}}{x^4}$

$\therefore$ Limit exist when

$a-b=0,c=0,1=\frac{a}{6}$

$\therefore b=6$ $\therefore a=6$

$\therefore a=6,b=6,c=0$