Limit of an expression converging towards the exponent?

Question

When does the limit of an expression converge towards its exponent?

$\lim_{x\to\infty} x^k \to k$

More precisely, what conditions does $k$ need to satisfy for this to be true and how can this be generalized to other cases such as:

$\lim_{x\to\infty} x^k \to k^2$

Answer 1

If $k=0$, then $\lim_{x\to\infty} x^k =1.$

If $k>0$, then $\lim_{x\to\infty} x^k =\infty.$

If $k<0$, then $\lim_{x\to\infty} x^k =0.$

Consequence: $\lim_{x\to\infty} x^k = k$ never holds !