In the last three paragraphs we discussed the meaning of the following limits: $$\begin{aligned} \lim_{x\to a} f(x) &= L, \qquad \lim_{x\uparrow a} f(x)=L, \qquad \lim_{x\downarrow a} f(x) = L\\ \lim_{x\to \infty} f(x) &= L ,\qquad\lim_{x\to -\infty} f(x) = L\text,\end{aligned}$$ where $L\in\mathbb R$. Just as the limit of a sequence ,these limits can be equal to $\pm\infty$. In this paragraph we will give the precise definitions belonging to this notion.

A sequence diverges to infinity if there does not exist an upper bound. This means that for every number$M$ the sequences grows larger than $M$ after a while. Similarly, the limit of $f(x)$ when $x$ approaches $a$ equals infinity if close to $x=a$ the values $f(x)$ are more than $M$. Put differently, there is a small interval around $x = a$ where the function takes values greater than $M$:

We denote this as $$\lim_{x\to a} f(x) = \infty$$

We denote this as $$\lim_{x\uparrow a} f(x) = \infty$$

We denote this as $$\lim_{x\downarrow a} f(x) = \infty$$

Finally we also have limits equal to $\pm\infty$. These definitions look a lot like the definitions of $\lim_{n\to\infty} a_n = \pm\infty$:

We denote this as $$\lim_{x\to \infty} f(x) = \infty$$

By replacing $f(x) > M$ by $f(x) < M$ in the five definitions you demand that the function for every $M$ for large $x$ values gets less than $M$. This gives the precise definitions of $$\begin{aligned} \lim_{x\to a} f(x) &= -\infty, \qquad \lim_{x\uparrow a} f(x)=-\infty, \qquad \lim_{x\downarrow a} f(x) = -\infty\\ \lim_{x\to \infty} f(x) &= -\infty ,\qquad\lim_{x\to -\infty} f(x) = -\infty\text.\end{aligned}$$

Note that if you know that $\lim_{x\to a} f(x) = \infty$, function $f$ cannot be continuous at $x = a$. The function cannot assume the value $\infty$ because this is a not a real number.