Limit(√n-100) ? n->♾

2 Answers
May 3, 2018

lim_(n->infty) f(n) = +infty

Explanation:

I'm not sure if you're looking for lim_(n->infty) sqrt(n-100) or lim_(n->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 (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->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(x) > k. Since f(x) is strictly increasing on x in (100, infty), this suffices to demonstrate that the limit is +infty.

May 3, 2018

The limit does not exist.

Explanation:

=>lim_(n->oo) {sqrt(n)-100} 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.