If the solve the problem
Question:

If $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}=\lim _{x \rightarrow k} \frac{x^{3}-k^{3}}{x^{2}-k^{2}}$, then $k$ is :

  1. $\frac{3}{8}$

  2. $\frac{3}{2}$

  3. $\frac{4}{3}$

  4. $\frac{8}{3}$


Correct Option: , 4

Solution:

$\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}=\lim _{x \rightarrow k} \frac{x^{3}-k^{3}}{x^{2}-k^{2}}$

$\Rightarrow \lim _{x \rightarrow 1}(x+1)\left(x^{2}+1\right)=\frac{k^{2}+k^{2}+k^{2}}{2 k}$

$\Rightarrow \mathrm{k}=8 / 3$

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