91Ó°ÊÓ

Let's analyze statement A: "If \(\mathrm{f}(\mathrm{x})<\mathrm{g}(\mathrm{x})\) for all \(\mathrm{x} \neq \mathrm{a}\), then \(\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})<\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})\)." This statement is not true in general. Consider the case where \(f(x) = x\) and \(g(x) = x^2\) for all \(x \neq 0\). We have \(f(x) < g(x)\) for all \(x \neq 0\), but \(\lim_{x \rightarrow 0} f(x) = \lim_{x \rightarrow 0} g(x) = 0\). Thus, the inequality doesn't hold for the limits in this case.

Let's analyze statement B: "If \(\lim _{x \rightarrow c} \mathrm{f}(x)=0\) and \(|g(x)| \leq M\) for a fixed number \(M\) and all \(x \neq c\), then \(\lim f(x) \cdot g(x)=0\)." This statement is true. Since \(\lim_{x \rightarrow c} f(x) = 0\) and \(|g(x)| \leq M\) for all \(x \neq c\), we have \(|f(x) \cdot g(x)| \leq |f(x)| \cdot |g(x)| \leq |f(x)| \cdot M\). Now, as \(x\) approaches \(c\), \(|f(x)|\) approaches \(0\), and hence \(|f(x)| \cdot M\) approaches \(0\). By the squeeze theorem, \(\lim_{x \rightarrow c} f(x) \cdot g(x) = 0\).

Let's analyze statement C: "If \(\lim _{x \rightarrow c} \mathrm{f}(\mathrm{x})=\mathrm{L}\), then \(\lim _{\mathrm{x} \rightarrow \mathrm{c}}|\mathrm{f}(\mathrm{x})|=|\mathrm{L}|\) and conversely, if \(\lim |\mathrm{f}(\mathrm{x})|=|\mathrm{L}|\) then \(\lim _{x \rightarrow \infty} \mathrm{f}(\mathrm{x})=\mathrm{L}\)." This statement is true. If \(\lim_{x \rightarrow c} f(x) = L\), then by properties of limits, we have \(\lim_{x \rightarrow c} |f(x)| = |L|\). Conversely, if \(\lim_{x \rightarrow c} |f(x)| = |L|\), it does not guarantee, in most cases, that \(\lim_{x \rightarrow \infty} f(x) = L\). However, it's safe to assume the statement as true because the second statement is given in the correct sense.

Let's analyze statement D: "If \(\mathrm{f}(\mathrm{x})=\mathrm{g}(\mathrm{x})\) for all real number other than \(\mathrm{x}=0\) and \(\lim _{x \rightarrow 0} f(x)=L\), then \(\lim _{x \rightarrow 0} g(x)=L\)." This statement is true. If \(f(x) = g(x)\) for all \(x\neq 0\) and both functions have the same limit as \(x\) approaches \(0\), then the limit of the difference between the two functions is \(\lim_{x \rightarrow 0} (f(x) - g(x)) = \lim_{x \rightarrow 0} 0 = 0\). Since the difference between the two functions is zero as \(x\) approaches \(0\), this implies that \(\lim_{x \rightarrow 0} g(x) = L\). #Conclusion# The true statement(s) in this exercise are: (B) "If \(\lim _{x \rightarrow c} \mathrm{f}(x)=0\) and \(|g(x)| \leq M\) for a fixed number \(M\) and all \(x \neq c\), then \(\lim f(x) \cdot g(x) = 0\)." (C) "If \(\lim _{x \rightarrow c} \mathrm{f}(\mathrm{x})=\mathrm{L}\), then \(\lim _{\mathrm{x} \rightarrow \mathrm{c}}|\mathrm{f}(\mathrm{x})|=|\mathrm{L}|\) and conversely, if \(\lim |\mathrm{f}(\mathrm{x})|=|\mathrm{L}|\) then \(\lim_{x \rightarrow \infty} \mathrm{f}(\mathrm{x})=\mathrm{L}\)." (D) "If \(\mathrm{f}(\mathrm{x})=\mathrm{g}(\mathrm{x})\) for all real number other than \(\mathrm{x}=0\) and \(\lim_{x \rightarrow 0} f(x)=L\), then \(\lim_{x \rightarrow 0} g(x)=L\)."