The Chemistry Maths Book, Second Edition

98 Chapter 4Differentiation

Point a.The function has a finite discontinuityatx 1 = 1 a. For example, the function

is discontinuous atx 1 = 10 where its value is −1. However, if we approach the point

x 1 = 10 from the right,f(x)approaches the value +1.

Point b.The function has an infinite discontinuityatx 1 = 1 b. If we approach this point

from the left, the value of the function tends to −∞; if we approach from the right the

function tends to +∞. Infinity is not a number, and the function is not defined at

x 1 = 1 b. An example is 12 (x 1 − 1 1), discontinuous atx 1 = 11.

Point c.The function tends to infinity if we approach the pointx 1 = 1 cfrom either side;

for example, 12 x

2

atx 1 = 10.

In these three cases, the nature of the discontinuities is obvious from the graphs; they

are said to be essential discontinuities. In some cases however, the discontinuity

is not obvious from the graph. For example, the functionf(x) 1 = 1 x 2 xhas constant

value equal to 1 for all values ofx 1 ≠ 10 , but it is not defined atx 1 = 10 because 0 2 0 is

indeterminate and has no meaning. Such a discontinuity is said to be removable. If we

redefine the function such thatf 1 = 1 x 2 xwhenx 1 ≠ 10 andf(x) 1 = 11 whenx 1 = 10 , then

the function becomes continuous for every value of x; the discontinuity has been

removed with no change to the function except at an isolated point.

0 Exercises 4–6

4.4 Limits

Consider the rational function

which is continuous for all values of xexceptx 1 = 12. Table 4.1 shows that, whereas both

the numerator and the denominator go to zero atx 1 = 12 , their ratioapproaches the

valuey 1 = 14 asx 1 → 12 from both sides:

Table 4.1 Values ofy 1 = 1 (x

2

1 − 1 4) 2 (x 1 − 1 2)

x 2.1 2.01 2.001 2.0001 = 1.9999 1.999 1.99 1.9

y 4.1 4.01 4.001 4.0001 = 3.9999 3.999 3.99 3.9

lim

x

x

x

→

−

−

















=

2

2

4

2

4

y

x

x

=

−

−

2

4

2

fx

xx

xx

()=

+>

−≤











10

10

if

if