Limits part 2: Functions: All kinds of limits
Limits at infinity
Just as one can study the long-term behaviour of sequences, one can also do that for functions. For example, the following holds: \[ \lim_{x\to\infty} \frac{1}{x}=0\text.\] The precise definition is:
The limit at infinity of a function \(f : \mathbb R\to\mathbb R\) is \(L\) if for every number \(\epsilon > 0\) there exists a number \(N\) such that if \(x > N\), then \(|f(x) - L|<\epsilon\).
This is denoted by \(\lim_{x\to\infty} f(x) = L\). Compare this with the definition of a limit of a sequence:
The limit of a sequence \((a_n)_{n=1}^\infty\) is \(L\) if for every number \(\epsilon>0\) there is a number \(N\) such that if \(n>N\), then \(|a_n - L|<\epsilon\).
There is a subtle difference between these two definitions: in the definition of \(\lim_{x\to\infty}f(x)=L\) must for all real numbers \(x>N\) hold that \(|f(x) - L|<\epsilon\) and in the definition of \(\lim_{n\to\infty}a_n = L\) we consider only natural numbers \(n > N\). In the following example, this difference is emphasized:
Consider the function \[ f(x) = \sin(2\pi x)\text. \] The period of this function is \(1\), so: \(f(0) = f(1) = f(2) = \dots = 0\). Thus, when we look at the sequence \( a_n = \sin(2\pi n)\), we see that \(a_n = 0\) all \(n\). So \(\lim_{n\to\infty} a_n = 0\), but the limit \(\lim_{x\to\infty} f(x)\) does not exist because the function oscillates between \(-1\) and \(1\).
Similarly, we define \(\lim_{x\to-\infty} f(x)\).
The limit at minus infinity of a function \(f : \mathbb R\to\mathbb R\) is \(L\) if for every number \(\epsilon > 0\) there exists a number\(N\) such that if \(x < N\) then \(|f(x) - L|<\epsilon\).
\[ \lim_{x\to-\infty} \frac{1}{x}=0 \]
