VietDao29

Homework Helper

- 1,417

- 1

This is one of the examples in the book I find really hard to understand.

[tex]\lim_{x \to 1} \left( \frac{a}{1 - x ^ a} - \frac{b}{1 - x ^ b} \right) \mbox{, ab \neq 0}[/tex]

Here is what the book says:

In some neighbourhood of x = 1, [itex]\alpha \neq 0[/itex]you have:

[tex]1 - x ^ \alpha = -\alpha (x - 1) - \frac{\alpha (\alpha - 1)}{2} (x - 1) ^ 2 + o((x - 1) ^ 2)[/tex]

This is Taylor's series. I understand this.

And the book continues:

[tex]\frac{a}{1 - x ^ a} - \frac{b}{1 - x ^ b} = \frac{a(1 - x ^ b) - b(1 - x ^ a)}{(1 - x^a)(1 - x ^ b)} = \frac{a - b}{2} + o((x - 1) ^ 2)[/tex]

This equation troubles me. I don't understand how they get it.

So they conclude:

[tex]\lim_{x \to 1} \left( \frac{a}{1 - x ^ a} - \frac{b}{1 - x ^ b} \right) = \frac{a - b}{2} \mbox{, ab \neq 0}[/tex]

Any help would be appreciated.

Viet Dao,