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Q. $\displaystyle\lim _{x \rightarrow \infty}\left(\frac{x^{2}}{3 x-2}-\frac{x}{3}\right)=$

Limits and Derivatives

Solution:

Consider $\displaystyle\lim _{x \rightarrow \infty}\left[\frac{x^{2}}{3 x-2}-\frac{x}{3}\right]$
$=\displaystyle\lim _{x \rightarrow \infty}\left[\frac{3 x^{2}-x(3 x -2)}{3(3 x-2)}\right]$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{2 x}{3(3 x-2)}$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{2}{3} \frac{1}{\left(3-\frac{2}{x}\right)}=\frac{2}{9}$