Relationship between sequences and real functions

Sequences and real functions are closely linked: a sequence can be seen as the restriction to the natural numbers of a function of a real variable.

If $(a_n)$ arises from a function $f:\mathbb{R}_{>0}\to\mathbb{R}$ by restriction to the naturals, that is $a_n = f(n)$, then the limit of the sequence $a_n$ coincides with the limit as $x\to+\infty$ of the function, if the latter exists: $\lim_{x\to+\infty}f(x) = L \implies \lim_{n\to+\infty}a_n = L.$

The converse, in general, is false: $f(x) = \sin(\pi x)$ has $f(n) = 0$ for every $n\in\mathbb{N}$ (constant sequence, limit $0$), but $\lim_{x\to+\infty}f(x)$ does not exist because $f$ oscillates.

Example — Computing a limit of a sequence via a function

$a_n = n\sin(1/n)$. Consider $f(x) = x\sin(1/x)$ with $x\to+\infty$ and substitute $t = 1/x$ (as $x\to+\infty$ we have $t\to 0^+$): $f(x) = \dfrac{\sin t}{t} \to 1$. Hence $a_n\to 1$.

Links

Topics: Sequences
Concepts: Limit of a sequence
Skills: Computing limits