The Chemistry Maths Book, Second Edition

7.7 Approximate values and limits 217

and

1.221400 1 < 1 e

0.2

1 < 1 1.221415

The exact value is 1.221403.

0 Exercise 72

Limits

The MacLaurin series shows how a function behaves when the variable is very small.

Thusln(1 1 + 1 x) 1 ≈ 1 xwhen xis small enough, so that in the immediate vicinity ofx 1 = 10

the functiony 1 = 1 ln(1 1 + 1 x)can be approximated by the straight liney 1 = 1 x. Similarly,

when xis small enough,

Another way of expressing the same results is in terms of limits. Thus (Figure 7.3)

so that in the limitx 1 → 10 ,

(7.27)

Similarly,

(7.28)

More generally, the use of power series provides a systematic way of determining the

limit of the quotent of two functions,

whenf(x) 1 → 10 andg(x) 1 → 10 asx 1 → 1 a. It is not possible to substitutex 1 = 1 ain this

case because the result would be the indeterminate 0 2 0. However, if both functions

are expanded as Taylor series, by equation (7.24), we have

lim

()

xa()

fx

→ gx

lim

cos

x

x

x

→

−











=

0

2

11

2

lim

sin

x

x

→ x











=

0

1

sinx

x

xx

=−

!

• 
  

!

1 −

35

24



cosx

xx x

=−

!

• 
  

!

1 −≈−

24

1

2

24 2



sinxx

xx

=− x

!

• 
  

!

−≈

35

35



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y=

sin x

x

Figure 7.3