Question

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

• A. $2$
• B. $3$
• C. $4$
• D. $1$

Answer 1

Answer: (B)

$3$

Answer 2

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