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

Studdy Solution
Simplify the expression further and evaluate the limit:
$$= n^2 \cdot \frac{6}{n^2} \cdot \lim_{n \to \infty} \frac{1 - \frac{1}{6n}}{\sqrt{1 + \frac{1}{n^2}} + \sqrt{1 - \frac{5}{n^2} + \frac{1}{n^3}}}$$
As $n \to \infty$, the terms $\frac{1}{n^2}$, $\frac{5}{n^2}$, and $\frac{1}{n^3}$ approach zero:
$$= 6 \cdot \frac{1}{2} = 3$$
The value of the limit is:
$$\boxed{3}$$