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 are:

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

Some common limits include:

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

Function limits