Evaluate the limit. limit as x approaches 1 of (x^1/5 - 1)/(x-1)

Question

Evaluate the following limit: Limit as x approaches 1 of (x^1/5-1)/(x-1).

Answer 1

Here is how to find this limit using l'hopital's rule. Hope this helps!

Answer 2

(x^(1/5)-1)/(x-1)

take derivatives and plug in 1 for x

(x^.8)/5)/1= 1/5

Answer 3

The denominator of (x^1/5-1)/(x-1) can be factored. Use long division. Divide (x-1) by (x^1/5-1) to get x^4/5+x^3/5+x^2/5+x^1/5+1. Then, after factoring, we have (x^1/5-1)/((x^4/5+x^3/5+x^2/5+x^1/5+1)(x^1/5-1)). The (x^1/5-1)s cancel out, leaving 1/(x^4/5+x^3/5+x^2/5+x^1/5+1). Next, substitute 1 in for x. This results in 1/(1^4/5+1^3/5+1^2/5+1^1/5+1) = 1/(1+1+1+1+1) = 1/5. Hence, the limit as x approaches 1 of (x^1/5-1)/(x-1) is 1/5