How to solve this limit

Question

lim as x tends to positive infinity (x+√x)/(x²+x+1)

Answer 1

You can also use L'Hopital's rule:

 

limit as x→∞ of (x+√x)/(x²+x+1) = limit as x→∞ of [1+1/(2√x)]/(2x+1)] = 0

Answer 2

Since the limit is for x approaching infinity you only need to consider the term with the highest power in the numerator and the term with the highest power in the denominator.  The limit of the given expression is the same as the limit as x approaches infinity of x/x².  This simplifies to be the limit as x approaches infinity of 1/x.  Now we realize as x approaches infinity 1/x approaches 0.  The limit is 0