Evaluate lim x->1

Question

Please help. Thanks!

Answer 1

Answer is 4/3.

Here’s why. Let y=x-1 then x^(1/3)=(1+y)^(1/3)=1+y/3+... and x^(1/4)=(1+y)^(1/4)=1+y/4+...

So x^(1/3)-1=y/3+... and x^(1/4)-1=y/4+...

The expression becomes limit as y➝0 of (y/3+...)/(y/4+...). When y is small this comes to (y/3)/(y/4)=4/3.