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  4. If displaystyle lim x arrow 0 ((1+x)1 / x-e+(1/2) e x/x2)=(11 e/k) then find k

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Q. If $\displaystyle\lim _{x \rightarrow 0} \frac{(1+x)^{1 / x}-e+\frac{1}{2} e x}{x^{2}}=\frac{11 e}{k}$ then find $k$

Limits and Derivatives

Solution:

Using expansion
$\displaystyle\lim _{x \rightarrow 0} \frac{\left(e-\frac{e x}{2}+\frac{11 e}{24} x^{2} \ldots \ldots\right)-e+\frac{e x}{2}}{x^{2}}=\frac{11 e}{24}$