the value of x^p+1-px-x+p/(x-1)²  if x tends to 1,p belongs to N

Dear Student,

Please find below the solution to the asked query :

$$\begin{gathered} \underset{\mathrm{x}\to 1}{\lim }\frac{{\mathrm{x}}^{\mathrm{p}+1}-\mathrm{px}-\mathrm{x}+\mathrm{p}}{{\left(\mathrm{x}-1\right)}^2} \\ \mathrm{Since} \mathrm{it} \mathrm{is} 0/0 \mathrm{form} \mathrm{after} \mathrm{putting} \mathrm{limit} \mathrm{so} \mathrm{using}, \mathrm{L}\text{'}\mathrm{hospital} \mathrm{rule} \\ =\underset{\mathrm{x}\to 1}{\lim }\frac{{\displaystyle \frac{\mathrm{d}}{\mathrm{dx}}}\left({\mathrm{x}}^{\mathrm{p}+1}-\mathrm{px}-\mathrm{x}+\mathrm{p}\right)}{{\displaystyle \frac{\mathrm{d}}{\mathrm{dx}}}{\left(\mathrm{x}-1\right)}^2} \\ =\underset{\mathrm{x}\to 1}{\lim }\frac{{\mathrm{x}}^{\mathrm{p}+1}\ln \left(\mathrm{p}+1\right)-\mathrm{p}-1}{2\left(\mathrm{x}-1\right)}= \frac{\ln 2-2}{0}= \infty \end{gathered}$$

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