Mathematical Methods for Physics and Engineering : A Comprehensive Guide

4.7 EVALUATION OF LIMITS

These can all be derived by straightforward application of Taylor’s theorem to

the expansion of a function aboutx=0.

4.7 Evaluation of limits

The idea of the limit of a functionf(x)asxapproaches a valueais fairly intuitive,

though a strict definition exists and is stated below. In many cases the limit of

the function asxapproachesawill be simply the valuef(a), but sometimes this

is not so. Firstly, the function may be undefined atx=a, as, for example, when

f(x)=

sinx

x

,

which takes the value 0/0atx= 0. However, the limit asxapproaches zero

does exist and can be evaluated as unity using l’Hopital’s rule below. Anotherˆ

possibility is that even iff(x) is defined atx=aits value may not be equal to the

limiting value limx→af(x). This can occur for a discontinuous function at a point

of discontinuity. The strict definition of a limit is thatiflimx→af(x)=lthen

for any numberhowever small, it must be possible to find a numberηsuch that

|f(x)−l|<whenever|x−a|<η.In other words, asxbecomes arbitrarily close to

a,f(x) becomes arbitrarily close to its limit,l. To remove any ambiguity, it should

be stated that, in general, the numberηwill depend on bothand the form off(x).

The following observations are often useful in finding the limit of a function.

(i) A limit may be±∞. For example asx→0, 1/x^2 →∞.

(ii) A limit may be approached from below or above and the value may be

different in each case. For example consider the functionf(x)=tanx.Asxtends

toπ/2frombelowf(x)→∞, but if the limit is approached from above then

f(x)→−∞. Another way of writing this is

lim

x→π 2 −

tanx=∞, lim

x→π 2 +

tanx=−∞.

(iii) It may ease the evaluation of limits if the function under consideration is

split into a sum, product or quotient. Provided that in each case a limit exists, the

rules for evaluating such limits are as follows.

(a) lim

x→a

{f(x)+g(x)}= lim

x→a

f(x) + lim

x→a

g(x).

(b) lim

x→a

{f(x)g(x)}= lim

x→a

f(x) lim

x→a

g(x).

(c) lim

x→a

f(x)

g(x)

=

limx→af(x)

limx→ag(x)

, provided that

the numerator and denominator are

not both equal to zero or infinity.

Examples of cases (a)–(c) are discussed below.