Question

What is the lim of (((1/x)-(1/2))/x-2) , as x approaches 2

Answer 1

I assume the denominator is $x-2$.

$\lim_{x\to2}\dfrac{\dfrac{1}{x}-\dfrac{1}{2}}{x-2}=\\ \lim_{x\to2}\dfrac{\dfrac{2}{2x}-\dfrac{x}{2x}}{x-2}=\\ \lim_{x\to2}\dfrac{\dfrac{2-x}{2x}}{x-2}=\\ \lim_{x\to2}\dfrac{2-x}{2x(x-2)}}=\\ \lim_{x\to2}-\dfrac{x-2}{2x(x-2)}}=\\ \lim_{x\to2}-\dfrac{1}{2x}}=\\ -\dfrac{1}{2\cdot2}=\\ -\dfrac{1}{4}=$