§3.2 Using derivatives to evaluate limits (continued)

Summary

Functions $f$ and $g$ satisfy $f(3) = g(3) = 0$, $f'(3) = g'(3) = 0$, $f''(3) = -2$, and $g''(3) = 1$. Then $\lim_{x \to 3}\frac{f(x)}{g(x)}$ equals

For $R(x) = \frac{3(x-a)(x-b)}{5(x-a)(x-c)}$ with $a$, $b$, $c$ distinct, the correct asymptote inventory is

In the sense that $\lim_{x \to \infty}\frac{f(x)}{g(x)} = 0$ means $g$ dominates $f$, which statement is correct?

With the same functions $f$ and $g$ as the double L'Hopital item, compute $\lim_{x \to 3}\frac{f(x)}{g(x)}$ numerically.

Evaluate $\lim_{x \to 0^+}x^{2x}$, given that $\ln(x^{2x}) = 2x\ln(x) \to 0$.

For every positive integer $n$, the function $e^x$ dominates $x^n$ in the sense that $\lim_{x \to \infty}\frac{x^n}{e^x} = 0$.