How to solve lim_(x->1) (x^(1/5)-1)/(x^(1/2)-1) ?

1 Answer
Oct 21, 2017

lim_(x->1) (x^(1/5)-1)/(x^(1/2)-1) = 2/5

Explanation:

Let t = x^(1/10)

Then:

lim_(x->1) (x^(1/5)-1)/(x^(1/2)-1) = lim_(t->1) (t^2-1)/(t^5-1)

color(white)(lim_(x->1) (x^(1/5)-1)/(x^(1/2)-1)) = lim_(t->1) (color(red)(cancel(color(black)((t-1))))(t+1))/(color(red)(cancel(color(black)((t-1))))(t^4+t^3+t^2+t+1))

color(white)(lim_(x->1) (x^(1/5)-1)/(x^(1/2)-1)) = lim_(t->1) (t+1)/(t^4+t^3+t^2+t+1)

color(white)(lim_(x->1) (x^(1/5)-1)/(x^(1/2)-1)) = 2/5

graph{(x^(1/5)-1)/(x^(1/2)-1) [-0.656, 1.844, -0.15, 1.1]}