a) Determine if the statement "If $a, b \in \mathbb{R} \setminus \{0\}$, then $\ln(ab) = \ln a + \ln b$" is true or false. b) Determine if the statement "If $f: A \subseteq \mathbb{R} \rightarrow \mathbb{R}$ is a function such that $(c, \infty) \subseteq A$ for some $c \in \mathbb{R}$ and $\lim_{x\to\infty} f(x) = 0$, then $f(x) = 0$ for some $x \in A$" is true or false.

1. Problem Description

a) Determine if the statement "If $a, b \in \mathbb{R} \setminus \{0\}$, then $\ln(ab) = \ln a + \ln b$" is true or false.

b) Determine if the statement "If $f: A \subseteq \mathbb{R} \rightarrow \mathbb{R}$ is a function such that $(c, \infty) \subseteq A$ for some $c \in \mathbb{R}$ and $\lim_{x\to\infty} f(x) = 0$, then $f(x) = 0$ for some $x \in A$" is true or false.

2. Solution Steps

a) The statement "If $a, b \in \mathbb{R} \setminus \{0\}$, then $\ln(ab) = \ln a + \ln b$" is false.

The domain of the natural logarithm function $\ln(x)$ is $x > 0$.

The statement $\ln(ab) = \ln a + \ln b$ is true if $a>0$ and $b>0$.

However, if $a < 0$ and $b < 0$, then $ab > 0$, so $\ln(ab)$ is defined. But $\ln a$ and $\ln b$ are not defined.

For example, let $a = -1$ and $b = -1$. Then $ab = 1$, so $\ln(ab) = \ln(1) = 0$. However, $\ln(a) = \ln(-1)$ and $\ln(b) = \ln(-1)$ are not real numbers.

More generally, $\ln(ab) = \ln|a| + \ln|b|$ if $a, b \neq 0$.

However, $\ln a$ and $\ln b$ may not exist.

b) The statement "If $f: A \subseteq \mathbb{R} \rightarrow \mathbb{R}$ is a function such that $(c, \infty) \subseteq A$ for some $c \in \mathbb{R}$ and $\lim_{x\to\infty} f(x) = 0$, then $f(x) = 0$ for some $x \in A$" is false.

Consider the function $f(x) = \frac{1}{x}$ defined on $A = [1, \infty)$. Then $(1, \infty) \subset A$, and $\lim_{x\to\infty} f(x) = \lim_{x\to\infty} \frac{1}{x} = 0$. However, $f(x) = \frac{1}{x} > 0$ for all $x \in [1, \infty)$. Therefore, there is no $x \in A$ such that $f(x) = 0$.

3. Final Answer

a) False

b) False