Question8. Calculate the value of the following limit: $$\lim _{n \rightarrow \infty} \sqrt{n^{4}+n^{2}}-\sqrt{n^{4}-5 n^{2}+n}$$

Studdy Solution
Divide each term by $n^2$ in the numerator and simplify:
$$= \lim_{n \to \infty} \frac{6 - \frac{1}{n}}{\sqrt{1 + \frac{1}{n^2}} + \sqrt{1 - \frac{5}{n^2} + \frac{1}{n^3}}}$$
As $n \to \infty$, $\frac{1}{n} \to 0$, $\frac{1}{n^2} \to 0$, and $\frac{1}{n^3} \to 0$. Therefore, the expression simplifies to:
$$= \frac{6 - 0}{\sqrt{1 + 0} + \sqrt{1 - 0 + 0}}$$
$$= \frac{6}{1 + 1}$$
$$= \frac{6}{2}$$
$$= 3$$
The value of the limit is:
$$\boxed{3}$$