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 method you should be taking is to take advantage of algebra, and in some cases L'Hospitals rule.
So for this question, we have $\lim_{x \to \infty} $ $ \frac{7x-3}{2x+2} $
We I'm sure you can see as x approached infinity we have the case of infinity/infinity so it is valid to use the rule. So  (7x-3)'=(7) and (2x+2)'=2 so we have $\lim_{x \to \infty} \frac{7}{2}  = 7/2$
Or as eluded to above, you can take advantage of the fact that $\lim_{x \to \infty} 7/x = 0 $
