# for a certain value of p limit x->-inf[(x^5+7x^4 +2)^p-x]=n is finite and non zero,then the value of p and n is

$\hspace{-0.6cm}Given l=\lim_{x\rightarrow \infty}\left[(x^5+7x^4+2)^p-x\right].\\\\ Now (x^5+7x^4+2)^p=x^{5p}\left[1+\left(\frac{7}{x}+\frac{2}{x^5}\right)\right]^p\\\\So we get x^{5p}\left[1+p\left(\frac{7}{x}\right)+\mathcal{O}\left(\frac{1}{x^2}\right)\right]\\\\So we get l=\lim_{x\rightarrow 0}\left[x^{5p}+\frac{7p\cdot x^{5p}}{x}+\mathcal{O}\left(\frac{1}{x^2}\right)-x\right]\\\\ So we get 5p=1\Rightarrow p=\frac{1}{5} and l=7p=\frac{7}{5}.$