Given find a value for n, which is a real number, such that the limit exists. (Such that the limit converges to some, presumably, real number.)

I'm omitting the work/algebra I've done to save time/space and because this all seems pretty clear to me.

It seems pretty obvious, for the case of that it doesn't converge, just by looking at the end behavior of any polynomial function with a positive exponent greater then 1, say 2 and the fact that an integral is the area under a curve. While x^1 is just a constant function and doesn't converge either.

For 0<n<1, we have some root function, which also doesn't converge. Again I'm just looking at the graph.

For -1<n<0, this is the inverse of a root. Which looks similar to the inverse of a polynomial. This range appears to converge on the interval (0,1] but diverge on [1,+ ] (I'm omitting work here.)

Since * Yet again I'll just reference the graph and conclude that this diverges over the given interval.

For n<-1, the function is just an inverse of a polynomial. They converge over the interval [1, + ). But all diverge over (0,1].

So assuming this is as clear as I think it is, a big assumption, and I haven't made any errors . There is no n satisfying the conditions.

So if i'm correct, and there is no n such that the limit exists/converges, could someone give me some hints as to how I can write this out in a way that isn't so bulky?

Thanks!