How can I prove the using definition of a limit?

Question

Given that lim_x→c f(x) = (√50) - 7.

I need to prove, using the definition of the limit, that there exists a δ > 0 such that f(x) > 0 for 0 < |x – c| < δ.

I have no idea how to do this when I am not given c or f(x). My book only gives examples on how to do this when the value of c is given and the function is also given.

Can anyone help?

Answer 1

Take $\delta$ such that $|f(x)-(\sqrt {5} -7)| <0.01$ for $0<|x-c| <\delta$. Then $0<|x-c| <\delta$ implies$|f(x)-(\sqrt {5} -7)| <0.01$ which implies $f(x) >\sqrt {5} -7-0.01 >0$ because $\sqrt {50} >7.01$.

Answer 2

1) Note $a:=\sqrt{50}-7 >0$.

$50 -49=(\sqrt{50}-7)(\sqrt{50}+7) >0$; Since the second factor is positive: $a>0$.

2) $\lim_{x \rightarrow c }f(x)= a (>0)$, i.e.

for $\epsilon >0$ there is a $\delta >0$ s.t.

$|x-c|\lt \delta$ implies $|f(x)-a|<\epsilon$.

Then

$-\epsilon +a <f(x)<\epsilon +a$.

3) Choose $\epsilon = a/2 (>0)$.

There exists a $\delta$ s.t.

$|x-c|<\delta$ implies

$0< -\epsilon +a= a/2 <f(x)$.