How to solve this limit (without hopital rule)

Question

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

Answer 1

Look at the numerator and denominator separately.

 

x + √x = √x (√x + 1), which approaches √x √x = x for large x, since √x + 1 approaches √x for large x.

 

x² + x + 1 approaches x² for large x.

 

Then for large x, the ratio approaches x/x² = 1/x, which approaches 0 as x→∞.

 

The limit is zero.  L'Hopital's rule also gives 0 for the limit.

Answer 2

I suggest you divide numerator and denominator through by the highest power of x that is present. That is a standard technique to see clearly what the limit will be. Highest power x² so essentially you are multiplying the expression above by (1/x²)/(1/x²)

 

(1/x²)/(1/x² )  * (x+√x)/(x²+x+1) = (1/x + 1/x^(3/2))/(1 + 1/x + 1/x²)

 

This makes it easy to see that as x goes to positive infinity all terms go to 0 except the 1 in the denominator. This leaves you with (0 + 0)/(1 + 0 + 0) and I will leave it to you to determine what final answer that gives!