- #1

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## Homework Statement

[tex]\frac{x^5-a^5}{x^2-a^2}[/tex], where a is some constant.

## Homework Equations

## The Attempt at a Solution

I can't figure out how to do this with long division. With synthetic, I can get to [tex]\frac{a^4+a^3 x+a^2 x^2+a x^3+x^4}{a+x}[/tex]

Code:

```
x^3+xa^2+?
_______________
x^2-a^2 ) x^5-a^5
-x^5 + x^3a^2
---------------
0- a^5+x^3a^2
xa^4-x^3a^2
---------------
```

Hopefully it's possible to decipher my steps from that diagram, I don't know how to write long division in latex. I'm left with -a^5+xa^4, which doesn't go evenly into the divisor. I thought about writing [tex]\frac{-a^5+xa^4}{x^2-a^2}[/tex] in place of the ? mark, but the entire purpose of this is to take the limit as x->a, and I would be left with division by zero. Surely this isn't only solvable by synthetic division?

Edit: For prosperity's sake, the limit is (5a^3)/2