Infinity limit with radicals

Question

limx-->infty (√3x⁵-2x) - (√3x⁵-7x) 

 

*In both brackets, the square root is over the entire polynomial.*

 

I tries solving by first rationalizing, and then when it become one fraction, i divided the numerator by x and the denominator by x^5.  i have a feeling im doing something wrong, so this is where i stopped.

Answer 1

When you rationalize the numerator you should get 3x⁵-2x - (3x⁵-7x) which is 5x;

the denominator can be approximated by a constant times x^5/2 since on the way to infinity the linear terms

in the radicands are negligible) and thus the limit will be a constant/x^3/2 which is 0.