In the final part of the ratio test, divide the numerator and the denominator by n and take n to infinity which takes all of the constants to zero. The bottom simplifies to k*k for k times. The numerator simplifies to 1^(k) or 1.
Within the absolute value bars, disregarding the $\displaystyle x$ what you have are $\displaystyle k$th degree polynomials in the numerator and denominator in $\displaystyle n$, so we know the limit as $\displaystyle n\to\infty$ is the ratio of the leading coefficients, which is:
$\displaystyle \frac{1}{k^k}$
And so the limit of the entire expression is:
$\displaystyle \left|\frac{x}{k^k} \right|$