Limits part 2: Functions: All kinds of limits
Limits equal to infinity
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 \(M\):
The limit of a function \(f\) at a point \(a\) is \(\infty\) if for all \(M\) there exists \(\delta > 0\) such that for all \(x\) satisfying \(0<|x - a|<\delta\) it holds that \(f(x) > M\).
We denote this as \[\lim_{x\to a} f(x) = \infty\]
The left limit of \(f\) at a point \(a\) is \(\infty\) if for all \(M\) there exists \(\delta>0\) such that \(a-\delta < x < a\) implies \( f(x) > M\).
We denote this as \[\lim_{x\uparrow a} f(x) = \infty\]
The left limit of \(f\) to a point \(a\) is \(\infty\) if for all \(M\) there exists \(\delta>0\) such that \(a < x < a + \delta\) implies \( f(x) > M\).
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 \(\lim_{n\to\infty} a_n = \pm\infty\):
The limit of \(f\) at \(\infty\) equals \(\infty\) if for every \(M\) there exists \(N\) such that \(f(x) > M\) for all \(x > N\).
We denote this as \[\lim_{x\to \infty} f(x) = \infty\]
The limit of \(f\) at \(\infty\) equals \(-\infty\) if for every \(M\) there exists \(N\) such that \(f(x) < M\) for all \(x > N\).
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.
