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

Studdy Solution
Now, evaluate the limit as $n$ approaches infinity:
$$\lim _{n \rightarrow \infty} \frac{6 - \frac{1}{n}}{\sqrt{1 + \frac{1}{n^2}} + \sqrt{1 - \frac{5}{n^2} + \frac{1}{n^3}}}$$
As $n \to \infty$, the terms $\frac{1}{n^2}$, $\frac{5}{n^2}$, and $\frac{1}{n^3}$ approach zero, so:
$$\sqrt{1 + \frac{1}{n^2}} \to 1$$ $$\sqrt{1 - \frac{5}{n^2} + \frac{1}{n^3}} \to 1$$
Thus, the denominator approaches $1 + 1 = 2$.
The limit becomes:
$$\lim _{n \rightarrow \infty} \frac{6 - \frac{1}{n}}{2} = \frac{6}{2} = 3$$
The value of the limit is:
$$\boxed{3}$$