Page 221 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes [Dividing throughout by x]

Ê x x 2 ˆ
x 1+ Á + + .......... ˜
= Ë 2! 3! ¯
x

= Ê 1+ x + x 2 ˆ
Á Ë 2! 3! + .......... ˜ ¯

ˆ
Ê
x
e - 1 - 1
2
\ limlim e x = lim + Ê Á x x x x 2 .................... ˆ ˜
lim 1=
+
+ +
=
+
+ 1
˜
Á
x Æ 0 x Æ 0 x x x Æ 0 Ë xÆ 0 Ë ! 2 2 ! 3 ! ! 3 ¯ ¯
= 1 + 0 + 0 + ...... = 1
x
Thus, lim e - 1 = 1
xÆ0 x
Example 6:
Examine the behaviour of the function in each of the following:
(i) when x → 2, x → – ∞ and x → ∞
(ii) when x → 1, x → – ∞ and x → ∞

(iii) when x → + ∞

Show the behaviour by sketching graph, indicating the asymptotes of the function.
Solution:
Note that y is not defined in each of the above cases.

(i) LHL

RHL

Also

and

Note that there is a vertical asymptote at x = 2 and a horizontal asymptote to the
function.

The behaviour of the function is shown in Fig. 7.3.

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