limit

Limit

The limit of a sequence $(a_n{)}_{n\in \mathbb{N}}$ or a function $f$ describes the value that the sequence/function approaches as the index/argument grows (or approaches some point).

See also: convergence.

Limit arithmetic

Let $(a_n),(b_n)$ be convergent sequences with $a:={\lim }_{n\to \infty }{a}_n$ and $b:={\lim }_{n\to \infty }{b}_n$, and let $\lambda \in \mathbb{C}$. Then:

$$\begin{array}{rl}\underset{n\to \infty }{\lim }(a_n+b_n)& =a+b\\ \underset{n\to \infty }{\lim }(\lambda \cdot a_n)& =\lambda \cdot a\\ \underset{n\to \infty }{\lim }(a_n\cdot b_n)& =a\cdot b\\ \underset{n\to \infty }{\lim }\frac{a_n}{b_n}& =\frac{a}{b}\text{if }b\ne 0\end{array}$$

The same rules hold for function limits ${\lim }_{x\to c}f(x)$.

Proofs

Sum rule: Use the triangle inequality and “epsilon budgeting”:

$$|a_n+b_n-(a+b)|\le |a_n-a|+|b_n-b|<\frac{\epsilon }{2}+\frac{\epsilon }{2}=\epsilon$$

for $n$ large enough. Each term gets half the epsilon budget.

Product rule: Add and subtract $a{b}_n$:

$$|a_n{b}_n-ab|=|a_n{b}_n-a{b}_n+a{b}_n-ab|\le |b_n||a_n-a|+|a||b_n-b|$$

Since convergent sequences are bounded, $|b_n|\le C$ for some $C$. Both terms can be made arbitrarily small.

Scalar rule: Follows directly from $|\lambda a_n-\lambda a|=|\lambda ||a_n-a|$.

Quotient rule: Show $\frac{1}{b_n}\to \frac{1}{b}$ first (using $|b_n|>\frac{|b|}{2}$ for large $n$), then apply the product rule.

Arithmetic for definitely divergent sequences

If $a_n\to \infty$:

$a_n+b_n\to \infty$ when $b_n\to \infty$ or $b_n\to b\in \mathbb{R}$
$\frac{\alpha }{a_n}\to 0$ for any $\alpha \in \mathbb{R}$
$\alpha \cdot a_n\to \infty$ if $\alpha >0$, and $\to -\infty$ if $\alpha <0$

Indeterminate forms (no general rule):
$\infty -\infty$
$\frac{\infty }{\infty }$
$0\cdot \infty$

Example: $a_n=(-1{)}^n/n\to 0$ and $b_n=n\to \infty$, but $a_n\cdot b_n=(-1{)}^n$ diverges, so since $0\cdot \infty$, it is indeterminate.

See sequence, null sequence, convergence for some examples.