Consider $$\int_{2}^{2+\epsilon}\frac{1}{x^2-2^2}dx=\left.\frac{1}{4}\ln\left|\frac{x-2}{x+2}\right|\right|_2^{2+\epsilon}=\frac{1}{4}\left[\ln\left(\frac{\epsilon}{4+\epsilon}\right)-\lim\limits_{x\rightarrow 2}\ln\left(\frac{x-2}{x+2}\right)\right]$$ for $\epsilon>0$. The limit in the bracket does not exist. However, Mathematica gives finite values for the integrations:
NIntegrate[1/(x^2/2-2^2/2),{x,2,2.3}]
74.7506
NIntegrate[1/(x^2/2-2^2/2),{x,2,3}]
74.6751
How Mathematica compute these integrations? And why integration value is decreasing even when the integration range is increasing?