I'd know how to solve this for $k=0$ or $k=1$ for example, but I'm currently lost trying to prove the limit is zero for any non-negative integer $k$. I'd appreciate any hints!
2026-03-27 10:24:24.1774607064
$\lim_{x\to 0}x^a\log^k(x)$ where $a>0,k\in\mathbb N_0$
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$\lim_{x\to 0}x^a\log^k(x)$ is also equal to$$\lim_{x\to 0}\frac{(-1)^k (log(\frac{1}{x}))^k}{\frac{1}{x^a}}$$, which turns indeterminancy to infinity/infinity form.
There are now two approaches,First you may apply L'Hospitals rule to get the answer.
Secondly, by graphical method you may see that POLYNMIAL growth as "h"(substituted as 1/x is this case, thus making limit h tending towards infinity), is far greater than logarithmic growth as "h tends towards infinity", therefore denomitor is far greater than numerator thus making limit tending to zero