SECTION 5.1 Quick Survey of Limits 249
Definition. Letf be a function defined in a neighborhood ofa. If
limx→af(x)−f(a)
x−a=L,
we say thatf is differentiableat x=aand writef′(a) =L, calling
f′(a)thederivative of f ata.
In mathematical analysis we often encounter the notion of a se-
quence, which is nothing more than a function
f: { 0 , 1 , 2 ,...}−→R.It is customary to write the individual terms of a sequence
f(0), f(1), f(2),...as subscripted quantities, say, asa 0 , a 1 , a 2 ,....
Sequences may or may not have limits.Definition. Let(an)n≥ 0 be a sequence. We say that thelimitof the
sequence is the real number L ∈ R and write nlim→∞an = L, if given
> 0 there exists a real number N such that whenever n > N then
|an−L|< .
We shall begin a systematic study of sequences (and “series”) in the
next section.
Finally, we would like to give one more example of a limiting process:
that associated with the “Riemann integral.” Here we have a function
fdefined on the closed interval [a,b], and apartitionP of the interval
intonsubintervals
P : a=x 0 < x 1 < x 2 <···< xn=b.On each subinterval [xi− 1 ,xi] let
Mi=xmax
i− 1 <x<xi
f(x), mi=x min
i− 1 <x<xi
f(x).The upper Riemann sum relative to the above partition is the
sum