Originally posted by SwissGambit
Here's something I couldn't look up:
The limit, as x goes to 1, of...
(x - x^(n-1))/(1-x)
...is n-2, but why?
OK, here is as spiffy a solution as I could find.
Convert the equation
f(x) = (x-x^(n-1))/(1-x)
Let's take the denominator off for a bit, so it's easier to read.
Now we start adding and subtracting the same term from the numerator over and over to factor out x-1 from every term. This will create new terms, but we'll do the same trick on them until EVERY term has an (x-1) in it.
x(x^(n-2) - x^(n-3) + x^(n-3) - 1)
x(x^(n-3)(x-1) + x^(n-3) - x^(n-4) + x^(n-4) - 1)
x(x^(n-3)(x-1) + x^(n-4)(x-1) + x^(n-4) - 1)
Keep doing this until you have dropped x's exponent down to one and you're left with an x-1 at the end.
x(x^(n-3)(x-1) + x^(n-4)(x-1) + ... + x(x-1) + x-1)
Now add the denominator back in:
x((x^(n-3)(x-1) + x^(n-4)(x-1) + ... + x(x-1) + x-1))/(x-1)
Cancel out all the (x-1)'s:
x(x^(n-3) + x^(n-4) + ... + x + 1)
Multiply the outer x through:
f(x) = x^(n-2) + x^(n-3) + ... + x^2 + x
Now that the denominator is gone, there is no more divide by 0 problem. With x=1 in each term, and n-2 total terms, the limit of f(x) as x approaches 1 is n-2