proof of limit using delta-epsilon definition

Question

Let $h=f+g$ with $f,\,g$ functions of domain $\Bbb R$. Prove if $\lim_{x\to\infty}f(x)=\infty$ and $\lim_{x\to\infty}g(x)\in\Bbb R$ then $\lim_{x\to\infty}h(x)=\infty$. Use the definition of limit, without limit laws or other theoems.

Could anyone share me an insight how to approach this question? and possibly how to prove this with proper format?

Answer 1

Given any number $M>0$, we have to show that there exists $a>0$, such that $h(x)>M$ for all $x>a$.

Now as $\lim_{x\to\infty}g(x)=L\in\Bbb R$, there exists $A$ such that $g(x)\in(L-1,L+1)$ for all $x\ge A$.

Also since $f(x)$ diverges to $\infty$ for $x\to\infty$, there exists $B$ such that $f(x)\ge M-L+1$ for all $x\ge B$.

Take $a=\max\{A,B\}$. Then for all $x\ge a$, $h(x)=f(x)+g(x)\ge (M-L+1)+(L-1)=M$.

Answer 2

Hint:

Simply write down the definition of the limits for $f$ and $g$. They both give you a lower bound on $f$ and $g$, and by a mere addition you get alower bound on $f+g$.

$$\forall M:\exists N':\forall x>N':f(x)>M\\\forall\epsilon:\exists N'':\forall x>N'':g(x)>G-\epsilon.$$ Then take $\epsilon=\dfrac1M$ and $N'''=\max(N',N'')$ and you get $$\forall M:\exists N''':\forall x>N''':f(x)+g(x)>M+G-\frac1M.$$