Understanding Engineering Mathematics

So asx→∞,

ex

xn

→∞–i.e.exis ‘stronger’ thanxn; for any positive integern.Sofor

example:
xn
ex

=xne−x→0asx→∞

In fact, it is not difficult to extend the above proof to apply for any value ofn, not
necessarily integer. This limit is very useful in Laplace transforms and should be thoroughly
understood (➤502).

Exercise on 14.3

Evaluate the following limits

(i) lim

x→ 0

sin 2x

x

(ii) lim

x→ 3

x^3 − 27

x− 3

(iii) lim

x→∞

2 x^3 −x

e^3 x

Answer

(i) 2 (ii) 27 (iii) 0

14.4 Continuity

We can now use the idea of a limit to put the concept of continuity into mathematical
form. If we focus on a particular pointx=aon the curve of a functionf(x), then clearly
if the curve is continuous at that point then we would want the limit atx=ato be the
same ‘from both sides’ and to be equal tof(a). Only in this way can the two parts of the
curve on either side ofx=a‘join up’ without leaving a hole in the curve. We express
this formally in the following definition:
A functionf(x)iscontinuous atx=aif limx→af(x)=f(a).
The graphs of continuous functions are continuous curves, and some examples of contin-
uous and discontinuous curves are illustrated in Figure 14.5.

y

0 x

y

0

0 0

y y

x x = a x

x = a x

Continuous

and

smooth

Continuous

but not

differentiable at x = a

Finite

discontinuity

at x = a

Infinite

discontinuity

at x = a

a

Figure 14.5Continuous and discontinuous curves.