Understanding Engineering Mathematics

Problem 14.2

Evaluatef.x/=x^2 Y 2 xY3forx= 1 .1, 1.01, 1.001, 1.0001. What do you

thinkf.x/approaches asxgets closer and closer to 1?

We obtain the values in the table below

xf(x)

1.1 6.41

1.01 6.0401

1.001 6.004001

1.0001 6.00040001

So asxgets closer and closer to 1,f(x)gets closer and closer to 6 – we say that the limit
asxtends to 1 is 6. Similar results are obtained if we consider values ofxless than and
increasingly closer to 1. The results get increasingly closer to 6. This is also the value of
f(x)atx=1 of course. Graphically, the situation is illustrated in Figure 14.1.

− 2 − 1 0

3

6

y y = x (^2) + 2 x + 3
1 x
Figure 14.1The graph ofy=x^2 + 2 x+3.
Problem 14.3
Evaluate the functionf.x/=
x^2 − 1
x− 1
forx= 1 .2, 1.02, 1.002,
1.0002 – what do you think is the limit asxtends to 1?
We obtain the table below
xf(x)
1.2 2.2
1.02 2.02
1.002 2.002
1.0002 2.0002