Limit as x approaches 1 for (x^2017 - 1)/x-1

Question

 I need to find the limit of this function NOT using L'hospital's rule. It is asking as x approaches 1 

Answer 1

Evaluate limit at x=0.999999 and at x=1.000001

 

since x=1 is the restriction. 

Answer 2

Use the property that

 

aⁿ - bⁿ = (a - b)(a^n-1 + ba^n-2 + b²a^n-3 + ... + b^n-2a + b^n-1)

 

a = x

n = 2017

b = 1

 

x²⁰¹⁷ - 1 = (x - 1)(x²⁰¹⁶ + 1x²⁰¹⁵ + 1²x²⁰¹⁴ + ... + 1²⁰¹⁵x + 1²⁰¹⁶

 

x²⁰¹⁷ - 1 = (x - 1)(x²⁰¹⁶ + x²⁰¹⁵ + x²⁰¹⁴ + ... + x + 1)

 

(x²⁰¹⁷ - 1) / (x - 1) = [(x - 1) / (x -1)] (x²⁰¹⁶ + x²⁰¹⁵ + x²⁰¹⁴ + ... + x + 1)

 

(x²⁰¹⁷ - 1) / (x - 1) = (x²⁰¹⁶ + x²⁰¹⁵ + x²⁰¹⁴ + ... + x + 1)

 

Now let x→1.

 

1²⁰¹⁶ = 1²⁰¹⁵ = 1²⁰¹⁴ = ... = 1¹ = 1

 

Note that the right hand side is just

 

1 + 1 + 1 + ... + 1 + 1 = 2017(1) = 2017

 

Then

 

liim_x→1 [(x²⁰¹⁷ - 1) / (x - 1)] = 2017

 

This is the same limit you get if you apply L'Hopital's rule.