Prove that lim[z->1] (z^x - 1)/(z^y - 1) = x/y

[Roger.Robert]

\(\displaystyle \displaystyle \lim _{z\rightarrow1}\frac{z^x-1}{z^y-1}=\frac{x}{y}\)

is true for x,y real numbers. How do i prove that this is true?

 

[stapel]

When i was solving this exercise:

\(\displaystyle \displaystyle \lim _{x\rightarrow1} \; \frac{x^4 - 1}{x^3 - 1}\)


which is equal to 4/3, and i noticed that the soultion seems to be the division between exponents. I tried with hundred of other values and this
(this is the function in python
from decimal import Decimal
l=lambda x,y: Decimal(Decimal((1-10**-16))**Decimal(x))-1)/(Decimal(Decimal((1-10**-16))**Decimal(y))-1)
)
\(\displaystyle \displaystyle \lim _{z\rightarrow1}\frac{z^x-1}{z^y-1}=\frac{x}{y}\)

seems to be true for all real numbers. My problem is how do i prove that this is true?

Hint: Do the long division of x^4 - 1 and x^3 - 1 by x - 1. What do you get? What happens, after cancellation, when you evaluate at x = 1?