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#.
