§3.2 Using derivatives to evaluate limits (continued)

Summary

Let $f$ and $g$ be differentiable functions about which the following information is known: $f(3) = g(3) = 0$, $f'(3) = g'(3) = 0$, $f''(3) = -2$, and $g''(3) = 1$. Let a new function $h$ be given by the rule $h(x) = \frac{f(x)}{g(x)}$. On the same set of axes, sketch possible graphs of $f$ and $g$ near $x = 3$, and use the provided information to determine the value of

.

Provide explanation to support your conclusion.

Find all vertical and horizontal asymptotes of the function

, where $a$, $b$, and $c$ are distinct, arbitrary constants. In addition, state all values of $x$ for which $R$ is not continuous. Sketch a possible graph of $R$, clearly labeling the values of $a$, $b$, and $c$.

Consider the function $g(x) = x^{2x}$, which is defined for all $x \gt 0$. Observe that $\lim_{x \to 0^+} g(x)$ is indeterminate due to its form of $0^0$. (Think about how we know that $0^k = 0$ for all $k \gt 0$, while $b^0 = 1$ for all $b \ne 0$, but that neither rule can apply to $0^0$.) - Let $h(x) = \ln(g(x))$. Explain why $h(x) = 2x \ln(x)$. - Next, explain why it is equivalent to write $h(x) = \frac{2\ln(x)}{\frac{1}{x}}$. - Use L'Hôpital's Rule and your work in (b) to compute $\lim_{x \to 0^+} h(x)$. - Based on the value of $\lim_{x \to 0^+} h(x)$, determine $\lim_{x \to 0^+} g(x)$.

Recall we say that function $g$ dominates function $f$ provided that $\lim_{x \to \infty} f(x) = \infty$, $\lim_{x \to \infty} g(x) = \infty$, and $\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$. - Which function dominates the other: $\ln(x)$ or $\sqrt{x}$? - Which function dominates the other: $\ln(x)$ or $\sqrt[n]{x}$? ($n$ can be any positive integer) - Explain why $e^x$ will dominate any polynomial function. - Explain why $x^n$ will dominate $\ln(x)$ for any positive integer $n$. - Give any example of two nonlinear functions such that neither dominates the other.