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Can anyone help me understand this please?
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,
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,