Cambridge Additional Mathematics

336 Introduction to differential calculus (Chapter 13)

In practice we do not need to graph functions each time to determine limits, and most can be found

algebraically.

Example 1 Self Tutor

Evaluate: a lim

x! 2

x^2 b lim

x! 0

x^2 +3x

x

c lim

x! 3

x^2 ¡ 9

x¡ 3

a x^2 can be made as close as we like to 4 by makingxsufficiently close to 2.

) lim

x! 2

x^2 =4.

b lim

x! 0

x^2 +3x

x

= lim

x! 0

x(x+3)

x

= lim

x! 0

(x+3) since x 6 =0

=3

c lim

x! 3

x^2 ¡ 9

x¡ 3

= lim

x! 3

(x+ 3)(x¡3)

x¡ 3

= lim

x! 3

(x+3) since x 6 =3

=6

EXERCISE 13A

1 Evaluate:

a lim

x! 3

(x+4) b lim

x!¡ 1

(5¡ 2 x) c lim

x! 4

(3x¡1)

d lim

x! 2

(5x^2 ¡ 3 x+2) e lim

h! 0

h^2 (1¡h) f lim

x! 0

(x^2 +5)

2 Evaluate:

a lim

x! 0

5 b lim

h! 2

7 c lim

x! 0

c, ca constant

3 Evaluate:

a lim

x! 1

x^2 ¡ 3 x

x

b lim

h! 2

h^2 +5h

h

c lim

x! 0

x¡ 1

x+1

d lim

x! 0

x

x

4 Evaluate the following limits:

a lim

x! 0

x^2 ¡ 3 x

x

b lim

x! 0

x^2 +5x

x

c lim

x! 0

2 x^2 ¡x

x

d lim

h! 0

2 h^2 +6h

h

e lim

h! 0

3 h^2 ¡ 4 h

h

f lim

h! 0

h^3 ¡ 8 h

h

g lim

x! 1

x^2 ¡x

x¡ 1

h lim

x! 2

x^2 ¡ 2 x

x¡ 2

i lim

x! 3

x^2 ¡x¡ 6

x¡ 3

Arateis a comparison between two quantities with different units.

We often judge performances by rates. For example:

² Sir Donald Bradman’s average batting rate at Test cricket level was 99 : 94 runs per innings.

² Michael Jordan’s average basketball scoring rate was 20 : 0 points per game.

² Rangi’s average typing rate is 63 words per minutewith an error rate of 2 : 3 errors per page.

B Rates of change

1

1

1

1

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\336CamAdd_13.cdr Friday, 4 April 2014 5:23:43 PM BRIAN