Evaluate the following limits: _ x 0 [(1-x)^5-1 (1-x)^3-1 ] - Vidyadip

Question

Evaluate the following limits:
$\lim _{x \rightarrow 0}\left[\frac{(1-x)^5-1}{(1-x)^3-1}\right]$

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Answer

$\lim _{x \rightarrow 0}\left[\frac{(1-x)^5-1}{(1-x)^3-1}\right]$
Put $1-x=y$
As $x \rightarrow 0, y \rightarrow 1$
$ =\frac{\lim _{y \rightarrow 1} \frac{y^5-1^5}{y-1}}{\lim _{y \rightarrow 1} \frac{y^3-1^3}{y-1}}$
$=\frac{5(1)^4}{3(1)^2}$
$\ldots\left[\lim _{x \rightarrow a} \frac{x^{\mathrm{n}}-\mathrm{a}^{\mathrm{n}}}{x-\mathrm{a}}=\mathrm{na}^{\mathrm{n}-1}\right]$
$=\frac{5}{3}$

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