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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#

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]}

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

How to solve #lim_(x-&gt;1) (x^(1/5)-1)/(x^(1/2)-1) ?#

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