Let $\lim _{x \rightarrow 1}\left[\frac{x^x-a x+a-1}{(x-1)^2}\right]=f(a)$. Then the value of $f(4)$

Question

Let $\lim _{x \rightarrow 1}\left[\frac{x^x-a x+a-1}{(x-1)^2}\right]=f(a)$. Then the value of $f(4)$ is $\_\_\_\_$ .

Professor PC Thomas Classes · Answer

Putting $x=1+h$, we get \ $ \begin{aligned} & f(a)=\lim _{h ightarrow 0} \frac{(1+h)^a-a(1+h)+a-1}{h^2} \ & =\lim _{h ightarrow 0} \frac{\left(1+a h \frac{a(a-1)}{2!} h^2+\ldots \ldotsight)-a-a h+a-1}{h} \ & =\frac{a(a-1)}{2} \end{aligned} $ \ Therefore $f(4)=6$.