# Prove $\lim\limits_{x\to\frac52}\frac1{4x-8}=0.5$ via delta epsilon

prove $$\lim\limits_{x\to\frac{5}{2}}\frac{1}{4x-8}=0.5$$ via delta epsilon I have the following

$$|\frac{1}{4x-8}-\frac{1}{2}|<\epsilon$$

$$|\frac{5-2x}{4x-8}|<\epsilon$$ which I got after simplification

But I want to have $$|x-\frac{5}{2}|$$ somewhere but I can't seem to be able to get that

Guide:

Dividing the numerator and denominator by $$2$$ and since $$|a|=|-a|$$, we have \begin{align} \left|\frac{5-2x}{4x-8}\right| = \left|\frac{x-\frac52}{2x-4}\right| \end{align}

Choose a small $$\delta$$ to control the magnitude of $$\frac1{|2x-4|}$$ and you should be able to prove it.