# How do you find the limit #(x+x^(1/2)+x^(1/3))/(x^(2/3)+x^(1/4))# as #x->oo#?

##### 2 Answers

#### Explanation:

The greatest power of

Another way to approach this is to factor the greatest power from the numerator and denominator:

#lim_(xrarroo)(x+x^(1/2)+x^(1/3))/(x^(2/3)+x^(1/4))=lim_(xrarroo)(x(1+x^(-1/2)+x^(-2/3)))/(x^(2/3)(1+x^(-5/12))#

#=lim_(xrarroo)(x^(1/3)(1+x^(-1/2)+x^(-2/3)))/(1+x^(-5/12))#

#=lim_(xrarroo)x^(1/3)*lim_(xrarroo)(1+x^(-1/2)+x^(-2/3))/(1+x^(-5/12))#

Note that for

#=lim_(xrarroo)x^(1/3)*(1+0+0)/(1+0)#

#=lim_(xrarroo)x^(1/3)#

#=oo#

A very slightly different approach.

#### Explanation:

We will write a quotient that is equivalent, but whose denominator does not go to infinity.

Factor out of both the top and bottom, the greatest power of

# = (x^(8/12)(x^(1/3)+1/x^(2/12)+1/x^(4/12)))/(x^(8/12)(1+1/x^(9/12))#

# = (x^(1/2)+1/x^(1/6)+1/x^(1/3))/(1+x^(3/4))#