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  4. If lim x arrow 2 ( tan (x-2) ⋅(x2+(k-2) x-2 k)/(x2-4 x+4))=5, then the value of k is

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Q. If $\lim _{x \rightarrow 2} \frac{\tan (x-2) \cdot\left(x^{2}+(k-2) x-2 k\right)}{\left(x^{2}-4 x+4\right)}=5$, then the value of $k$ is ____

Limits and Derivatives

Solution:

We have $\displaystyle\lim _{x \rightarrow 2} \frac{\tan (x-2) \cdot\left(x^{2}+(k-2) x-2 k\right)}{(x-2)^{2}}=5$
$\Rightarrow \displaystyle\lim _{x \rightarrow 2} \frac{\tan (x-2)}{x-2} \cdot \frac{\left(x^{2}+(k-2) x-2 k\right)}{x-2}=5$
$\Rightarrow \displaystyle \lim _{x \rightarrow 2} \frac{x^{2}+(k-2) x-2 k}{x-2}=5$
$\Rightarrow x^{2}+(k-2) x-2 k=(x-2)(x+\lambda)$
For the value of limit to be $5, \lambda=3$.