Evaluate lim (x --> 1) (x^1000 - 1) / (x - 1)

Question

Evaluate lim (x --> 1) (x¹⁰⁰⁰ - 1) / (x - 1)

 

I tried using the conjugate but that didn't do anything for me

 

 

#77 on p183

ANSWER:   1,000

Answer 1

By direct substitution, we get 0/0, so L'Hopital's Rule applies.  

 

Applying the rule yields the desired result.  

 

 

Answer 2

It is true that L'Hopital's Rule can quickly give you the answer. For the student, however, who is not acquainted with this Rule, it is recommended that you divide out.

Note that x-1 is a factor of   x^1000 - 1 by the Remainder - Factor Theorem. Use synthetic division  and you will obtain    (x^1000 - 1) / (x - 1) = x^999 + x ^998 + x ^997 + . . . . .. + x + 1

lim (x --> 1) = 1 + 1 + 1 + .......+1 = 1000