Asymptotes

Investigate and graph the following functions:

$y=\frac{x^4}{{\left(1+x\right)}^3}$

Investigate and graph the following functions :

$y=1+x^2-\frac{x^4}{2}$

Find the asymptotes of the following curves :

$y=2x-\arccos \frac{1}{x}$.

Find the asymptotes of the following curves :

$y=\ln \left(4-x^2\right)$;

Find the asymptotes of the following curves :

$y=x+\frac{\left(\sin x\right)}{x}$

Find the asymptotes of the following curves :

$y=x\arctan x$

Find the asymptotes of the following curves :

$y=\frac{x^2-6x+3}{x-3}$

Show that the points of inflection of the curve $y=x\sin x$ lie on the curve $y^2\left(4+x^2\right)=4x^2$.

Proceeding directly from the definition, ascertain that the straight line $y=2x+1$ is an asymptote of the curve $y=\frac{2x^4+x^3+1}{x^3}$.