This question from limits has been bothering me. Could anyone help me.
$\lim_{x\rightarrow 0}{(a^x + b^x +c^x/3) ^{1/x}}$
This question from limits has been bothering me. Could anyone help me.
$\lim_{x\rightarrow 0}{(a^x + b^x +c^x/3) ^{1/x}}$
If we take exponentials and logarithms, we will get $\exp\left(\lim_{x\rightarrow 0}\frac{\log(a^x + b^x + c^x) - \log(3)}{x}\right)$ Let us examine the inner limit. We can see that the inner limit is of the indeterminant form $\frac{0}{0}$. We apply L'Hopital's rule: $\overset{H}{=}\lim_{x\rightarrow 0}\frac{a^x\log a + b^x\log b + c^x\log c}{a^x + b^x + c^x}$ Can you finish off the limit from here? Don't forget to exponentiate the final result.