Evaluate the following Limits : _ x 1 [x^3-1 x^2+5 x-6 ] - Vidyadip

Question

Evaluate the following Limits : $\lim _{x \rightarrow 1}\left[\frac{x^3-1}{x^2+5 x-6}\right]$

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Answer

$ \lim _{x \rightarrow 1} \frac{x^3-1}{x^2+5 x-6}$
$ =\lim _{x \rightarrow 1} \frac{(x-1)\left(x^2+x+1\right)}{(x-1)(x+6)}$
$ =\lim _{x \rightarrow 1} \frac{x^2+x+1}{x+6} \ldots\left[\begin{array}{l}\text { As } x \rightarrow 1, x \neq 1 \\ \therefore x-1 \neq 0\end{array}\right]$
$ =\frac{(1)^2+1+1}{1+6}$
$ =\frac{3}{7} \text { }$

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