# Thread: limits of the sum of functions at a real number

1. ## limits of the sum of functions at a real number

Can someone give me a clue as to how to solve the following:
Assume:$\displaystyle f,g\to \mathbb{R}$, with $\displaystyle D\subseteq \mathbb{R}$ and a is an accumulation point of D.
Prove that if $\displaystyle \lim_{x\to a}f(x)=+\infty$ and $\displaystyle \lim_{x\to a}g(x)=L$, then $\displaystyle \lim_{x\to a}(f+g)(x)=+\infty$.

also, I was wondering how to show that:
$\displaystyle \lim_{x\to a}(\frac{g}{f})(x)=0$

2. Originally Posted by dannyboycurtis
Can someone give me a clue as to how to solve the following:
Assume:$\displaystyle f,g\to \mathbb{R}$, with $\displaystyle D\subseteq \mathbb{R}$ and a is an accumulation point of D.
Prove that if $\displaystyle \lim_{x\to a}f(x)=+\infty$ and $\displaystyle \lim_{x\to a}g(x)=L$, then $\displaystyle \lim_{x\to a}(f+g)(x)=+\infty$.

$\displaystyle \color{red}\mbox{As}\,\,(f+g)(x)=f(x)+g(x)\,\,\mbo x{, this follows at once from arithmetic of limits}$

also, I was wondering how to show that:
$\displaystyle \lim_{x\to a}(\frac{g}{f})(x)=0$

$\displaystyle \color{red}\mbox{It's easy to show that something bounded times something going to zero}$ $\displaystyle \color{red}\mbox{ must go it all to zero}$

Tonio

3. Its not clear that (f+g)(x) = f(x)+g(x) because one ofthem tends to infinity, the limit laws dont hold in this case do they?

4. still not sure what to do in this case

5. Originally Posted by dannyboycurtis
Its not clear that (f+g)(x) = f(x)+g(x) because one ofthem tends to infinity, the limit laws dont hold in this case do they?
Yes they do, since the other limit is finite, but you can also show it directly: is it true that

$\displaystyle \forall M\in\mathbb{R}\,\,\exists r\in\mathbb{R}\,\, s.t.\,\,f(x)+g(x)>M\,\,for\,\,\,x>r?$

Yes, it is true, since as $\displaystyle \lim_{x\rightarrow\infty}g(x)=L$ then $\displaystyle g(x)$ is bounded, so $\displaystyle -a<g(x)<a$ for some
$\displaystyle a\in\mathbb{R}$ and for all $\displaystyle x$ , so we need to check that $\displaystyle f(x)+g(x)>f(x)-a$ , and $\displaystyle f(x)-a$ can be made as large as we want as $\displaystyle x\rightarrow\infty$ since $\displaystyle \lim_{x\rightarrow\infty}f(x)=\infty!!$

Tonio