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  4. displaystyle limx → 1[((4x/x2-x-1)-(1-3x+x2/1-x3))-1+3((x4-1/x3-x-1))] is

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Q. $\displaystyle \lim_{x \to 1}$$\left[\left(\frac{4x}{x^{2}-x^{-1}}-\frac{1-3x+x^{2}}{1-x^{3}}\right)^{-1}+3\left(\frac{x^{4}-1}{x^{3}-x^{-1}}\right)\right]$ is

Limits and Derivatives

Solution:

Given limit can be written as,
$\displaystyle \lim_{x \to 1}$$\left[\left(\frac{4x}{x^{3}-1}-\frac{1-3x+x^{2}}{1-x^{3}}\right)^{-1}+\frac{3x\left(x^{4}-1\right)}{x^{4}-1}\right]$
$=\displaystyle \lim_{x \to 1}$$\left[\left(\frac{4x+1-3x+x^{2}}{x^{3}-1}\right)^{-1}+3x\right]$
$=\displaystyle \lim_{x \to 1}$$\left[\frac{x^{3}-1}{x^{2}+x+1}+3x\right]$
$=\displaystyle \lim_{x \to 1}(x - 1 + 3x) = 3$