Unit - 5

Expansion of Functions and Indeterminate forms

Question and answer

1. Then

Proof:-

Here f(x) = sin hx.

By Maclaurin’s expansion,

(1)

By equation (1) we get,

2.     . Then

Proof:-

Here f(x) = cos hx

By Maclaurin’s expansion

(1)

By equation (1)

3.     f(x) = tan hx

Proof:

Here f(x) = tan hx

By Maclaurin’s series expansion,

… (1)

By equation (1)

4.     then

Proof:-

Here f(x) = log (1 + x)

By Maclaurin’s series expansion,

… (1)

By equation (1)

5.    

Solution:

Here f(x) = log (1 + sin x)

By Maclaurin’s Theorem,

… (1)

……..

equation (1) becomes,

6.     Expand in power of (x – 3)

Solution:

Let

Here a = 3

Now by Taylor’s series expansion,

… (1)

equation (1) becomes.

7.     Using Taylors series method expand

in powers of (x + 2)

Solution:

Here

a = -2

By Taylors series,

… (1)

Since

,, …..

Thus equation (1) becomes

8.     Expand in ascending powers of x.

Solution:

Here

i.e.

Here h = -2

By Taylors series,

… (1)

equation (1) becomes,

Thus

9.     Expand in powers of x using Taylor’s theorem,

Solution:

Here

i.e.

Here

h = 2

By Taylors series

… (1)

By equation (1)

10. Evaluate

Solution:

Let

…

By L – Hospital rule