How Do I solves these types of limits such as $\lim_{x\to\infty}\frac{7x-3}{2x+2}$ I guess the best question I have is how does my strategy change when I get a limit such as 
$$\lim_{x\to\infty}\dfrac{7x-3}{2x+2}$$
What is essential to have as an understanding to solve these problems? Help welcomed.  

The problem of the limit having been nicely explained, if you already know simple approximations, you could do something more to answer two questions : what is the limit and how is it approached ?
For more generality, let me consider the case of $$A=\frac{ax+b}{cx+d}$$ what, as said in answers can write $$A=\frac{a+\frac bx}{c+\frac dx}=\frac{a(1+\frac b{ax})}{c(1+\frac d{cx})}$$ Now, if you know or remember that, for small values of $y$, $\frac 1{1+y}\approx 1-y$, then $$A\approx \frac a c(1+\frac b{ax})(1-\frac d{cx})=\frac{a}{c}+\frac{b c-a d}{c^2 x}$$ So, depending on the sign of $(bc-ad)$, the limit will be approached from above an below.
Hoping this helps.
