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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

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
 

Grade:12

1 Answers

jagdish singh singh
173 Points
5 years ago
\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 $l=7p=\frac{7}{5}.$

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