How to prove $\lim_{x \to \infty} x^{(x+1)}-(x+1)^x = \infty$ I try to prove this by using L'Hospital Rule but it doesn't work.
I know it is infinity from wolframalpha but I don't know how to prove it.

Use the estimate:
$$n^{n+1}-(n+1)^n>n^n, n>3 \iff n^n(n-1)>(n+1)^n,n>3 \iff \\
n-1>\left(1+\frac 1n\right)^n, n>3.$$
