Using derivatives to evaluate limits

Evaluate each of the following limits. If you use L'Hôpital's Rule, indicate where it was used, and be certain the limit is indeterminate before you apply it.

1. $\displaystyle \lim_{x \to 0} \frac{\ln(1 + x)}{x}$

2. $\displaystyle \lim_{x \to \pi} \frac{\cos(x)}{x}$

3. $\displaystyle \lim_{x \to 1} \frac{2 \ln(x)}{1-e^{x-1}}$

4. $\displaystyle \lim_{x \to 0} \frac{\sin(x) - x}{\cos(2x)-1}$

Evaluate each of the following limits. If you use L'Hôpital's Rule, indicate where it was used, and be certain its hypotheses are met before you apply it.

1. $\displaystyle \lim_{x \to \infty} \frac{x}{\ln(x)}$

2. $\displaystyle \lim_{x \to \infty} \frac{e^{x} + x}{2e^{x} + x^2}$

3. $\displaystyle \lim_{x \to 0^+} \frac{\ln(x)}{\frac{1}{x}}$

4. $\displaystyle \lim_{x \to \frac{\pi}{2}^-} \frac{\tan(x)}{x-\frac{\pi}{2}}$

5. $\displaystyle \lim_{x \to \infty} xe^{-x}$

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 $0^0$.)

1. Let $h(x) = \ln(g(x))$. Explain why $h(x) = 2x \ln(x)$.

2. Next, explain why it is equivalent to write $h(x) = \frac{2\ln(x)}{\frac{1}{x}}$.

3. Use L'Hôpital's Rule and your work in (b) to compute $\lim_{x \to 0^+} h(x)$.

4. 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 $\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$.

1. Which function dominates the other: $\ln(x)$ or $\sqrt{x}$?

2. Which function dominates the other: $\ln(x)$ or $\sqrt[n]{x}$? ($n$ can be any positive integer)

3. Explain why $e^x$ will dominate any polynomial function.

4. Explain why $x^n$ will dominate $\ln(x)$ for any positive integer $n$.

5. Give any example of two nonlinear functions such that neither dominates the other.