Hi, Im having trouble finding the limit as n goes to infiniti of ((n+1)(n+2))/(2n^2). Can someone please help me start this problem?
Thank you
$\displaystyle \lim_{n \to {\infty}}\frac{(n+1)(n+2)}{2n^2}=lim_{n \to {\infty}}\frac{n^2+3n+2}{2n^2}$...now direct substitution yields $\displaystyle \frac{0}{0}$...so you can use L'hopital's rule so $\displaystyle \lim_{n \to {\infty}}\frac{(n+1)(n+2)}{2n^2}=\lim_{n \to {\infty}}\frac{2n+3}{4n}$...once again the indeterminate form $\displaystyle \frac{0}{0}$..so L'hopitals rule again so $\displaystyle \lim_{n \to {\infty}}\frac{(n+1)(n+2)}{2n^2}=\lim_{n \to {\infty}}\frac{2}{4}=\frac{1}{2}$..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.
if this is what you did arbolis I am sorry I am just making it clear through LaTeX...ok you have $\displaystyle \lim_{n \to {\infty}}\frac{(n+1)(n+2)}{2n^2}=\lim_{n \to {\infty}}\frac{n^2+3n+2}{2n^2}=\lim_{n \to {\infty}}\frac{n^2+3n+2}{2n^2}\cdot\frac{\frac{1}{ n^2}}{\frac{1}{n^2}}$ which equals$\displaystyle \lim_{n \to {\infty}}\frac{1+\frac{3}{n}+\frac{2}{n}}{2}=\frac {1+0+0}{2}=\frac{1}{2}$
Hello,
I do think it's easier. This is a method learnt before l'Hôpital's rule. The study of the degree can help :
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.
but what if you were setting up the formal proof that the derivative of $\displaystyle sin(x)$ is $\displaystyle cos(x)$ and you ran into $\displaystyle \lim{\Delta{x} \to 0}\frac{1-cos(\Delta{x})}{\Delta{x}}$ or if you had $\displaystyle \lim_{t \to 0}\frac{\sqrt{1+t^2}-t}{t}$ what then...do you use L'hopitals or do you do something differnet...jus curious