Error of linear regression as a result of the coefficients
You have some inherent error in $m$, say that you predict $m=m_0+a$, where $m_0$ is the true value and where you can control the size of $a$ in terms of your error bounds on $m$. Then you get:

$$y=(m_0+a)x+b=m_0x+ax+b.$$

The $ax$ term is what will cause your error to grow as you increase/decrease $x$ toward $\pm\infty$. In other words, your predicted line will ultimately be infinitely far from the true line (as can be visualized by drawing two lines with slightly different slopes).