Question

$\underset{x\to 0}{\lim }{\left(\frac{(x+2\cos x{)}^3+2(x+2\cos x{)}^2+3\sin (x+2\cos x)}{(x+2{)}^3+2(x+2{)}^2+3\sin (x+2)}\right)}^{\frac{10}{x}}$ is equal to ___

Answer 1

Answer: 1

${\lim }_{x\to 10}{\left(\frac{(x+2\cos x{)}^3+2(x+2\cos x{)}^2+3\sin (x+2\cos x)}{(x+2{)}^3+2(x+2{)}^2+3\sin (x+2)}\right)}^x$
Form $1^{\infty }$
$=e^{{\lim }_{x\to 0}\left[\left(\frac{(x+2\cos x{)}^3+2(x+2\cos x{)}^2+3\sin (x+2\cos x)}{(x+2{)}^3+2(x+2{)}^2+3\sin (x+2)}\right)-1\right]\times \frac{100}{x}}$
$=e^{{\lim }_{x\to 0}\left[\frac{100}{x}\left(\frac{(x+2\cos x{)}^3+2(x+2\cos x{)}^2+3\sin (x+2\cos x)-\left((x+2{)}^3+2(x+2{)}^2+3\sin (x+2)\right)}{(x+2{)}^3+2(x+2{)}^2+3\sin (x+2)}\right)\right]}$
$=e^{{\lim }_{x\to 0}\frac{100}{x}\left[\left(\frac{(x+2\cos x{)}^3+(x+2{)}^3+2(x+2\cos x{)}^2-2(x+2{)}^2+3\sin (x+2\cos x)-3\sin (x+2)}{8+8+3\sin {)}^2}\right)\right]}$
$=e^{\frac{100}{16+3{\sin }^2}{\lim }_{x\to 0}\frac{3(x+2\cos x{)}^2\times (1+2\sin x)-3(x+2{)}^2-4(x+2\cos x)}{x(1-2\sin x)-4(x+2)+3\cos (x+2\cos x)\times (1-2\sin x)-3\cos (x+2)}}$
$=e^{\frac{100}{16+3\sin 2}2}\left(\frac{12-3(4)+8\times 1-8+3\cos 2-3\cos 2}{1}\right)$
Using L'H rule.
$=e^o=1$