How do you find the limit of ((x^35)-1)/((x^6)-1) as x approaches 1

Please explain the steps/limit laws you've used.

Thanks! :)

We factor as follows:
limit of (x^35 - 1)/(x^6 - 1) as x approaches 1
= limit of [(x - 1)(x^34 + x^33 + ... + 1)]/[(x - 1)(x^5 + x^4 + ... + 1)] as x approaches 1
= limit of (x^34 + x^33 + ... + x + 1)/(x^5 + x^4 + ... + x + 1) as x approaches 1
= (1^34 + 1^33 + ... + 1^1 + 1^0)/(1^5 + 1^4 + ... + 1^1 + 1^0)
= 35/6

Note that since the numerator went from 1^0 to 1^34, there are 35 1's in the numerator.
Likewise since the denominator went from 1^0 to 1^5, there are 6 1's in the denominator.

You plug in numbers into the equation that are very close to 1 but not 1 and you'll probably see that this number is approaching some number and that number is the limit.