Q54 / 120 · VITEEE 2017 · Mathematics

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

  • A
    $\frac{1}{3}$
  • B
    $\frac{2}{3}$
  • C
    $\frac{-2}{3}$
  • Correct answer
    D
    $\frac{2}{9}$

Solution

Consider $\lim _{x \rightarrow \infty}\left[\frac{x^{2}}{3 x-2}-\frac{x}{3}\right]$ $$ \begin{array}{l} =\lim _{x \rightarrow \infty}\left[\frac{3 x^{2}-x(3 x-2)}{3(3 x-2)}\right] \\ =\lim _{x \rightarrow \infty} \frac{2 x}{3(3 x-2)}=\lim _{x \rightarrow \infty} \frac{2 x}{3 x\left[3-\frac{2}{x}\right]} \\ =\lim _{x \rightarrow \infty} \frac{2}{3} \frac{1}{\left(3-\frac{2}{x}\right)}=\frac{2}{3} \times \frac{1}{3-0}=\frac{2}{9} \end{array} $$

Hence, the correct answer is option: 'D'.

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