Q: Find lim as n tends to infinity of
n(a^(1/n)-1)
where a>0.
Could anyone give me a hint of where to start please?
Thanks.
In general, $\displaystyle f'(x)=\lim_{h\to 0}\frac{a^{x+h}-a^x}{h}=a^x\lim_{h\to 0}\frac{a^h-1}{h}$. Now let $\displaystyle t=a^h-1\Rightarrow h=\log_a(t+1)=\frac{\log(t+1)}{\log a}$ and when $\displaystyle h\to 0, t\to 0$ so $\displaystyle \lim_{h\to 0}\frac{a^h-1}{h}=\lim_{t\to 0}\frac{\log a}{\frac{1}{t}\log(t+1)}=\log a$.
Finally, $\displaystyle f'(x)=a^x\log a$.