Proof that √x does not tend to a limit as x approaches infinity

I am wanting to prove from the definition of a limit ( $∀ε>0 ∃K>0:∀x>K, |f(x)−l|$<ε) that $√x$ does not tend to a limit as $x$ approaches infinity.

So far I have tried to find a value of K such that $|f(x)−l|$<ε holds, and was planing on picking an $x$ bigger than K that would give $f(x)>l$, however I am having trouble finding a value for K.

Thanks for any suggestions

Answer

By definition, $\lim_{x\to\infty}f(x)=L$ if $\forall\epsilon\gt 0$ there exists $N\gt 0$ such that $x\gt N\implies |f(x)-L|\lt\epsilon$

$|f(x)-L|\lt\epsilon$ is equivalent to $L-\epsilon\lt f(x)\lt L+\epsilon$.

To show that the limit does not exist, we can show that for any $L$, there is a fixed $\epsilon$ for which the inequality cannot hold.

Since we are talking about the square root we can assume $L$ is positive. Let $\epsilon = 1$. Then it is sufficient to show that we can always find $x$ such that $\sqrt{x}\ge L+1$.

Let $x=max(N+1, (L+2)^2)$. Then $x\gt N$ and $\sqrt{x}=\sqrt{(L+2)^2}=L+2\gt L+1$ so $L$ cannot be the limit. Since this is true for all $L$, the square root does not converge to a limit as $x$ approaches infinity.