# Limit questions

• Feb 15th 2013, 07:33 AM
asilvester635
Limit questions
For (f) I know that the limit as x approaches 0 is 2 for f(x), and the limit as x approaches 0 for g(x) is 0
So is this how you put them together????

2(2) + 3(0)
4 + 0 = 4
• Feb 15th 2013, 08:10 AM
HallsofIvy
Re: Limit questions
Yes, there should be a theorem in your book that says that if $\displaystyle \lim_{x\to a} f(x)= F$ and $\displaystyle \lim_{x\to a}g(x)= G$, then $\displaystyle \lim_{x\to a}(f+ g)(x)= F+ G$.

There should also be theorems that say that, under the same hypotheses,
$\displaystyle \lim_{x\to a}(f- g)(x)= F- G$.
$\displaystyle \lim_{x\to a}(fg)(x)= FG$.
and, provided g(x) is not 0 in some neighborhood of a,
$\displaystyle \lim_{x\to a}(\frac{f}{g})(x)= \frac{F}{G}$.

There is another limit theorem that is not emphasised nearly enough:
If f(x)= g(x) for all x except a, the $\displaystyle \lim_{x\to a}f(x)= \lim_{x\to a}g(x)$
So that if, for example $\displaystyle f(x)= x^2$ for all x and $\displaystyle g(x)= x^2$ for all x except x= 1 and g(1)= 1000, then $\displaystyle \lim_{x\to 1}g(x)= \lim_{x\to 1}f(x)= 1$.