...now direct substitution yields ...so you can use L'hopital's rule so ...once again the indeterminate form ..so L'hopitals rule again so ..or if its polynomial with same highest degree exponents you jsut divide the coefficents..
Your expression is (n+1)(n+2)/2n^2. It's equal to (n^2+3n+2)/2n^2. When you have this, factorize by the variable of the major exponent. In this case it's n^2. So we get (n^2(1+3/n+2/n^2)/(n^2*(2)). Simplify the numerator with the denominator. We get (1+3/n+2/n^2)/2. As n tends to positive infinite, the expression tends to 1/2.
This is usally the method teached before to learn l'Hôpital rule.
limit at infinity :
if deg(numerator)>deg(denominator) -> tends to infinity (sign is to determine with the coefficient of the highest power)
if deg(denominator)>deg(numerator) -> tends to 0
if deg(denominator)=deg(numerator) -> tends to the quotient of the two coefficients corresponding to the highest powers.
First I look the expression to look if there is any indetermination. If there is one like in our example, mindly I do the quotient of the variable afected by the major degree taking in count the coefficients. If I have to answer on paper, I do all the steps.