Question

Why is m_{x \to \infty} \frac{x}{ln x}" alt="lim_{x \to \infty} \frac{x}{ln x}" align="absmiddle" class="latex-formula"> equal to infinity, but $\lim_{x \to -\infty}\frac{2^x}{x^2}$ is equal to 0?

Answer 1

The first limit is infinity since $x>\ln(x)$ for all $x>0$.

The second limit is zero since $2^x$ converges to 0 as $x\to-\infty$, while $x^2$ is very large and positive.