Define the term: $f$ does not converge to $-\infty$ at $x_{0}$

Question

i am looking anywhere and i don't find a good explanation about this particular topic.

Let f be a function define on a deleted neighbourhood of ${ x_0} $. Define the term: $f$ does not converge to $-\infty$ at ${ x_0} $ .

please i will be very happy if someone can explain it completely

Answer 1

If $f(x)$ does not tend to $- \infty$ (get very negative) as $x$ tends to $x_0$, then it does not go to $-\infty$.

Formally,

$$\lim_{x \to x_0} f(x) \neq -\infty$$

where one should interpret $\neq$ as there exists some $L$ so that $|f(x)|<L$ (f is bounded) as $x \to x_0$, or $f(x)$ gets arbitrarily large, $f(x) \to \infty$ instead.

Answer 2

By definition $\lim_{x\to a} f(x)=-\infty$ iff for all $M$ there is a $\delta>0$ such that $0<|x-a|<\delta$ implies $f(x)<-M$.

Therefore $\lim_{x\to a} f(x)=-\infty$ does not hold iff there is an $M$ without such a $\delta>0$. In particular no $\delta={1\over n}$ does the job. Hence for each $n$ there is a point $x_n$ with $0<|x_n-a|<{1\over n}$ and at the same time $f(x_n)\geq M$.

In all we can say the following: The statement $\lim_{x\to a} f(x)=-\infty$ does not hold iff there is an $M$ and a sequence $x_n\to a$, whereby $x_n\ne a$ for all $n$, such that $f(x_n)\geq -M$.