# What is the limit of #sqrt(9x+x^2)/(x^4+7) # as x approaches infinity?

##### 1 Answer

#### Explanation:

We have the limit

#lim_(xrarroo)sqrt(9x+x^2)/(x^4+7)#

Factor out the largest-degreed terms from the numerator and denominator of the fraction.

#=lim_(xrarroo)sqrt(x^2(9/x+1))/(x^4(1+7/x^4))#

Note that the

#=lim_(xrarroo)(xsqrt(9/x+1))/(x^4(1+7/x^4))=lim_(xrarroo)(sqrt(9/x+1))/(x^3(1+7/x^4))#

When analyzing this as it goes to infinity, we see that

#=sqrt(0+1)/(oo(0+1))=1/oo=0#

There is also a more intuitive approach to limits of this type.

In the numerator, we have in a square root a polynomial of degree

In the denominator, the overpowering term is of degree

Since