Asymptotic density of k-almost primes

Landau's well known asymptotic formula $$N_k(x):=\ \mid\{n\leq x : \Omega(n)=k\}\mid \ \sim \left( \frac{x}{\log x} \right) \frac{(\log\log x)^{k-1}}{(k - 1)!}\ \ (x \rightarrow \infty),$$ which also holds for $$\pi_k(x):=\ \mid\{n\leq x : \omega(n)=k\}\mid,$$ is known to be fairly poor for $k > 1$, and when $k$ is allowed to tend to infinity with $x$, the study of $N_k(x)$ and $\pi_k(x)$ becomes very technical [1, Chapter II.6, $\S$ 6.1, p.200]. I hope to show that the method described below provides not only a more accurate approach, but rather increases in its asymptotic accuracy as $k$ tends to infinity.