Lim +- 0 (1+2^(1/x))/(1-2^(1/x))
Problem: Compute $\displaystyle L=\lim_{x\to0^+}\frac{1+2^{\frac{1}{x}}}{1-2^{\frac{1}{x}}}$
Solution: Let $\displaystyle z=\frac{1}{x}$ which transforms this into $\displaystyle L=\lim_{x\to0^+}\frac{1+2^{\frac{1}{x}}}{1-2^{\frac{1}{x}}}\overbrace{\longmapsto}^{z=\frac{1 }{x}}\lim_{z\to\infty}\frac{1+2^z}{1-2^z}$. Now, with a little manipulation $\displaystyle L=\lim_{z\to\infty}\frac{\frac{1}{2^z}+1}{\frac{1} {2^z}-1}$. Clearly $\displaystyle \frac{1}{2^z}\to0$ so that $\displaystyle L\to -1$.