# Limit(√n-100) ? n->♾

May 3, 2018

${\lim}_{n \to \infty} f \left(n\right) = + \infty$

#### Explanation:

I'm not sure if you're looking for ${\lim}_{n \to \infty} \sqrt{n - 100}$ or ${\lim}_{n \to \infty} \sqrt{n} - 100$, but in either case the answer will be the same.

When considering the limit of a function, we are only concerned with the greatest power of $n$ that exists. This can be tricky to determine when you're provided with a rational function (something like $\frac{{x}^{2} - 2}{{x}^{2} + 1}$), but that is not the case we're working with.

Notice that as we let $n$ get large, $\sqrt{n - 100}$ continues to get large with it. In fact, there is no finite limit for this function as $n \to \infty$, so we say the limit is $+ \infty$.

That is, you can pick any large number $k$ you like and you will always be able to find some $x$ such that $f \left(x\right) > k$. Since $f \left(x\right)$ is strictly increasing on $x \in \left(100 , \infty\right)$, this suffices to demonstrate that the limit is $+ \infty$.

May 3, 2018

The limit does not exist.

#### Explanation:

$\implies {\lim}_{n \to \infty} \left\{\sqrt{n} - 100\right\}$ tends to infinity

Square root of infinity is still infinite. Subtracting $100$ from infinity is still infinite. Therefore, the limit tends to infinity.

Hence, the limit DNE.