QuestionCalculate the value of the following limit: $$\lim _{n \rightarrow \infty} \sqrt{3 n^{2}+n}-\sqrt{3 n^{2}-5 n}$$

Studdy Solution
Evaluate the limit as $n \rightarrow \infty$:
As $n \rightarrow \infty$, $\frac{1}{n} \rightarrow 0$ and $\frac{5}{n} \rightarrow 0$, so:
$\sqrt{3 + \frac{1}{n}} \rightarrow \sqrt{3}$ $\sqrt{3 - \frac{5}{n}} \rightarrow \sqrt{3}$
Thus, the expression simplifies to:
$\lim _{n \rightarrow \infty} \frac{6}{\sqrt{3} + \sqrt{3}} = \frac{6}{2\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3}$
The value of the limit is:
$$\boxed{\sqrt{3}}$$