Limits

If a sequence \((a_n)\) converges to \(L \in \mathbb{R},\) the sequence is convergent and has a limit \(L.\)

\[L = \lim_{n\rightarrow \infty} a_n\]

The statement “\((a_n)\) converges to \(L\)” is represented as \(a_n \rightarrow L\) as \(n \rightarrow \infty .\)

The limit laws state that if \((a_n)\) and \((b_n)\) are convergent sequences then:

\[\lim_{n\rightarrow \infty} \left( a_n \pm b_n \right) = \lim_{n\rightarrow \infty} a_n \pm \lim_{n\rightarrow \infty} b_n\] \[\lim_{n\rightarrow \infty} \left( c \cdot a_n \right) = c \cdot \lim_{n\rightarrow \infty} a_n, \, c \in \mathbb{R}\] \[\lim_{n\rightarrow \infty} \left( a_n \cdot b_n \right) = \lim_{n\rightarrow \infty} a_n \cdot \lim_{n\rightarrow \infty} b_n\] \[\lim_{n\rightarrow \infty} \left( \frac{a_n}{b_n} \right) = \frac{\lim_{n\rightarrow \infty} a_n}{\lim_{n\rightarrow \infty} b_n} \,\, \mathrm{if} \,\, b_n \neq 0 \, \forall n \,\, \mathrm{and} \,\, \lim_{n\rightarrow \infty} b_n \neq 0\]

The convergence assumption matters: if \(a_n = n\) and \(b_n = -n\) then \(a_n + b_n \rightarrow 0\) even though neither \(\lim_{n\rightarrow \infty} a_n\) nor \(\lim_{n\rightarrow \infty} b_n\) exists.

Some common limits include:

\[\lim_{n\rightarrow \infty} \frac{1}{n} = 0\] \[\lim_{n\rightarrow \infty} c = c \, \, \forall c \in \mathbb{R}\] \[\lim_{n\rightarrow \infty} x^n = 0 \, \, \forall \, \lvert x\rvert < 1\] \[\lim_{n\rightarrow \infty} x^{\frac{1}{n}} = 1 \, \, \forall \, x > 0\]

Let \(X \subseteq \mathbb{R}\) be a subset of real numbers, \(f: X \rightarrow \mathbb{R}\) be a function and \(x_0 \in \mathbb{R}\) be a limit point of \(X,\) meaning that every open interval containing \(x_0\) contains a point of \(X\) other than \(x_0.\) The function \(f\) has the limit \(L \in \mathbb{R}\) as \(x \rightarrow x_0\) if and only if for every \(\varepsilon > 0\) there exists \(\delta > 0\) such that \(f(x)\) is \(\varepsilon\)-close to \(L\) for all \(x \in X\) with \(0 < \lvert x-x_0\rvert < \delta.\)

\[\lim_{x\rightarrow x_0} f(x) = L \Leftrightarrow \forall \varepsilon > 0 \, \exists \delta > 0 : \lvert f(x)-L \rvert \leq \varepsilon \, \forall x \in X, \, 0 < \lvert x-x_0\rvert < \delta\]

The value \(f(x_0),\) if it is defined at all, plays no part in the limit.