Evaluate the following limits: _ x 1 [ x^2+x x-2 x-1 ] - Vidyadip

Question

Evaluate the following limits:
$\lim _{x \rightarrow 1}\left[\frac{x^2+x \sqrt{x}-2}{x-1}\right]$

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Answer

$\lim _{x \rightarrow 1}\left[\frac{x^2+x \sqrt{x}-2}{x-1}\right]$
$=\lim _{x \rightarrow 1}\left[\frac{\left(x^2-1\right)+(x \sqrt{x}-1)}{x-1}\right]$
$=\lim _{x \rightarrow 1}\left[\frac{x^2-1}{x-1}+\frac{x^{\frac{3}{2}}-1}{x-1}\right] \ldots\left[\because x \sqrt{x}=x^1 \cdot x^{\frac{1}{2}}=x^{1+\frac{1}{2}}=x^{\frac{3}{2}}\right]$
$=\lim _{x \rightarrow 1}\left(\frac{x^2-1^2}{x-1}\right)+\lim _{x \rightarrow 1}\left(\frac{x^{\frac{3}{2}}-1^{\frac{3}{2}}}{x-1}\right) \quad \ldots\left[\because \lim _{x \rightarrow 2} \frac{x^n-\mathrm{a}^n}{x-\mathrm{a}}=\right.\text { na }\left.=2(1)^1+\frac{3}{2}(1)^{\frac{1}{2}}\right]$
$=2+\frac{3}{2}=\frac{7}{2}$

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