Answer
(a) $\displaystyle\lim_{x\rightarrow 0^{-}} f(x)=-1$
$\displaystyle\lim_{x\rightarrow 0^{+}} f(x)=1$
(b) $f(x)\rightarrow -1$ when $x\rightarrow 0^{-}$
$f(x)\rightarrow 1$ when $x\rightarrow 0^{+}$
Work Step by Step
We are given the function:
$f(x)=\dfrac{2^{1/x}-2^{-1/x}}{2^{1/x}+2^{-1/x}}$
(a) Compute $\displaystyle\lim_{x\rightarrow 0^{-}} f(x)$ and $\displaystyle\lim_{x\rightarrow 0^{+}} f(x)$:
$\dfrac{2^{1/-0.1}-2^{-1/-0.1}}{2^{1/-0.1}+2^{-1/-0.1}}\approx -0.99999809$
$\dfrac{2^{1/-0.01}-2^{-1/-0.01}}{2^{1/-0.01}+2^{-1/-0.01}}\approx -1$
$\dfrac{2^{1/-0.005}-2^{-1/-0.005}}{2^{1/-0.005}+2^{-1/-0.005}}\approx -1$
$\dfrac{2^{1/-0.001}-2^{-1/-0.001}}{2^{1/-0.001}+2^{-1/-0.001}}\approx -1$
$\dfrac{2^{1/0.001}-2^{-1/0.001}}{2^{1/0.001}+2^{-1/0.001}}\approx 1$
$\dfrac{2^{1/0.005}-2^{-1/0.005}}{2^{1/0.005}+2^{-1/0.005}}\approx 1$
$\dfrac{2^{1/0.01}-2^{-1/0.01}}{2^{1/0.01}+2^{-1/0.01}}\approx 1$
$\dfrac{2^{1/0.1}-2^{-1/0.1}}{2^{1/0.1}+2^{-1/0.1}}\approx 0.99999809$
We got:
$\displaystyle\lim_{x\rightarrow 0^{-}} f(x)=-1$ $\displaystyle\lim_{x\rightarrow 0^{+}} f(x)=1$
(b) Graph the function and zoom to describe its behaviour near $x=0$:
Notice that when $x$ approaches 0 from the left, the function's value approaches -1 and when $x$ approaches 0 from the right, the function's value approaches 1.
