I was trying to derive a general expression for the limit $$\large{y=\lim_{x\to a} f(x)^{g(x)}}$$ where $\lim_{ x \to a} f(x)=0$ and $\lim_{ x \to a} g(x)=\infty$ $$$$ I managed to reach till here: $$\ln (y)=\lim_{x\to a} g(x).\ln(f(x))$$
The result I'm supposed to reach is $$\LARGE{y=e^{\lim_{x\to a} \frac{f(x)}{g(x)}}}$$ $$$$ I would be grateful for any help. Many thanks!
PS. This was given to me by a friend. I'm not really sure if the question is correct.
This is not true. To see it let's consider a counterexample.
Let $$f(x)=x, g(x)=\frac1x\Rightarrow\\l_1=\lim_{x\rightarrow 0+}f(x)^{g(x)}=\lim_{x\rightarrow 0+}(x)^{\frac1x}=0$$
But $$l_2=e^{\lim_{x\rightarrow 0+}\frac{f(x)}{g(x)}}=e^{\lim_{x\rightarrow 0+}x^2}=e^0=1\neq0$$