Answer
The slope of the curve at $x=3$ is $-1/4$.
Work Step by Step
The slope $m$ of the tangent line of the curve $f(x)$ at $A(a,b)$ is also the slope of the curve at that point, which is calculated by $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$f(x)=\frac{1}{x-1}\hspace{1cm}x=3$$
To calculate the slope of the curve at $x=3$, we apply the above formula:
$$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
Here $a=x=3$
$$m=\lim_{h\to0}\frac{f(h+3)-f(3)}{h}$$
$$m=\lim_{h\to0}\frac{\frac{1}{h+3-1}-\frac{1}{3-1}}{h}=\lim_{h\to0}\frac{\frac{1}{h+2}-\frac{1}{2}}{h}=\lim_{h\to0}\frac{\frac{2-(h+2)}{2(h+2)}}{h}$$
$$m=\lim_{h\to0}\frac{-h}{2h(h+2)}=\lim_{h\to0}\frac{-1}{2(h+2)}$$
$$m=\frac{-1}{2(0+2)}=-\frac{1}{4}$$
The slope of the curve at $x=3$ is $-1/4$.
