we are given
[tex] \lim_{n \to \infty} \frac{6n^2+5}{3n^2} [/tex]
we can see that
degree of numerator = degree of denominator =2
so, we can divide both numerator and denominator by n^2
and we get
[tex] \lim_{n \to \infty} \frac{(6n^2+5)/n^2}{(3n^2/n^2)} [/tex]
[tex] \lim_{n \to \infty} \frac{(6n^2/n^2)+(5/n^2)}{(3n^2/n^2)} [/tex]
now, we can simplify it
[tex] \lim_{n \to \infty} \frac{6+(5/n^2)}{3} [/tex]
now, we can plug n=inf
[tex] = \frac{6+(5/(\infty))}{3} [/tex]
[tex] = \frac{6+0}{3} [/tex]
[tex] = \frac{6}{3} [/tex]
[tex] = 2 [/tex]...............Answer
[tex] \displaystyle
\lim_{n\to \infty} \dfrac{6n^2+5}{3n^2 }=\\\\
\lim_{n\to \infty} \dfrac{n^2\left(6+\dfrac{5}{n^2}\right)}{3n^2 }=\\\\
\lim_{n\to \infty} \dfrac{6+\dfrac{5}{n^2}}{3}=\\\\
\dfrac{6+0}{3}=2 [/tex]
