Show that the limit as x approaches zero for $\frac{2^{1/x} - 2^{-1/x}}{2^{1/x} + 2^{-1/x}}$ does not exist This problem was in one of the first chapters of a calculus text, so how would you go about solving this without applying L'Hôpital's rule?
I attempted factoring out $2^{1/x}$, as well as using u substitution for $2^{1/x}$.
By graphing, I can see that the left-side limit does not equal the right-side limit, but how else can I demonstrate this?

Quite simple with equivalents:
$$2^{1/x}\pm2^{-1/x}\sim_{0^+} 2^{1/x},\enspace\text{hence}\quad\frac{2^{1/x}-2^{-1/x}}{2^{1/x}+2^{-1/x}}\sim_{0^+}\frac{2^{1/x}}{2^{1/x}}=1,$$
while
$$2^{1/x}\pm2^{-1/x}\sim_{0^-}\pm 2^{-1/x},\enspace\text{hence}\quad\frac{2^{1/x}-2^{-1/x}}{2^{1/x}+2^{-1/x}}\sim_{0^-}\frac{-2^{-1/x}}{2^{-1/x}}=-1.$$
