I like l'hopitals rule for both of those equations. To use L'hopital's rule, we have to make sure they satisfy the condition that the functions at that point are of the indeterminate form 0/0 or inf/inf. These both look like 0/0, so we're good to go.
Now, since we've satisfied that condition, L'hopital's rule says that the limit of each function will be the same as if we take the derivative of the top and bottom.
So basically, if our function is f(x)/g(x), the rule says that the limit will be the same as f'(x)/g'(x), as long as that limit exists.
What that gives us is:
(-3x^2 + 8x - 3)/1 <= g(x) <= 2x/1
Evaluating at x=1 gives:
(-3 + 8 -3)/1 <= g(x) <= 2/1
2 <=g(x) <=2
So, by squeeze theorem, lim g(x) is 2.