Values of a limit with parameter

Question

Evaluate the values of the limit as the changes of $\lambda \in \mathbb{R}$ $$\lim\limits_{x \to +∞} e^{x-x^3+\lambda\ln(x)}f^{(n)}x$$

with $$f^{(n)}(x)=p_n(x)e^{x^3-x}$$ where $p_n(x)$ is a generic polynomial.

Answer 1

This is simply $$\lim_{x\to\infty}\exp\left(\not x-\not x^3+\lambda\ln(x)+\not x^3-\not x\right)p_n(x)\\ = \lim_{x\to\infty}x^\lambda p_n(x)$$ This limit can be evaluated given $p_n(x),\lambda$.