Answer
The graph is shown below.
(a) As $x\to0^+$, the graph approaches $0$.
(b) As $x\to\pm\infty$, the graph approaches $3/2$.
(c) As $x\to1$, the graph approaches $\infty$ from both the right and the left and as $x\to-1$, the graph approaches continuously a value of about $0.9449$.
Work Step by Step
$$y=\frac{3}{2}\Big(\frac{x}{x-1}\Big)^{2/3}$$
The graph is shown below.
(a) As $x\to0^+$, the graph approaches $0$.
Reason: $$\lim_{x\to0^+}\frac{3}{2}\Big(\frac{x}{x-1}\Big)^{2/3}=\frac{3}{2}\Big(\frac{0}{0-1}\Big)^{2/3}=\frac{3}{2}\times0^{2/3}=0$$
(b) As $x\to\pm\infty$, the graph approaches $3/2$.
Reason: $$A=\lim_{x\to\pm\infty}\frac{3}{2}\Big(\frac{x}{x-1}\Big)^{2/3}$$
Divide the numerator and denominator both by $x$, we have
$$A=\lim_{x\to\pm\infty}\frac{3}{2}\Big(\frac{1}{1-\frac{1}{x}}\Big)^{2/3}$$ $$A=\frac{3}{2}\Big(\frac{1}{1-0}\Big)^{2/3}=\frac{3}{2}\times1^{2/3}=\frac{3}{2}$$
(c) As $x\to1$, the graph approaches $\infty$ from both the right and the left and as $x\to-1$, the graph approaches continuously a value of about $0.9449$.
Reason: As $x\to1$, $$x-1\to0$$ $$\frac{x}{x-1}\to\infty$$ $$\frac{3}{2}\Big(\frac{x}{x-1}\Big)^{2/3}\to\infty$$
$$\lim_{x\to-1}\frac{3}{2}\Big(\frac{x}{x-1}\Big)^{2/3}=\frac{3}{2}\Big(\frac{-1}{-1-1}\Big)^{2/3}=\frac{3}{2}\Big(\frac{1}{2}\Big)^{2/3}\approx0.9449$$
