Question

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

Answer 1

`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]}