Question

Exploring Properties of Limits Find the limits of \(f, g,\) and \(f g\) as \(x \rightarrow c .\) (a) $$f(x)=\frac{1}{x}, \quad g(x)=x, \quad c=0$$ (b) $$f(x)=-\frac{2}{x^{3}}, \quad g(x)=4 x^{3}, \quad c=0$$

Short answer

\((a) \lim_{{x \to 0}} f(x) = \infty, \lim_{{x \to 0}} g(x) = 0, \lim_{{x \to 0}} fg(x) = undefined;\) \n \((b) \lim_{{x \to 0}} f(x) = - \infty, \lim_{{x \to 0}} g(x) = 0, \lim_{{x \to 0}} fg(x) = undefined\)

Step-by-step solution

Compute the limits for (a)

For \(f(x) = \frac{1}{x}\), as \(x\) approaches \(0\), \(f(x)\) approaches infinity: \n \[\lim_{{x \to 0}} f(x) = \lim_{{x \to 0}} \frac{1}{x} = \infty\]. For \(g(x) = x\), as \(x\) approaches \(0\), \(g(x)\) approaches \(0\): \[\lim_{{x \to 0}} g(x) = \lim_{{x \to 0}} x = 0\]. To find the limit of the product \(fg\), apply the limit product rule: \[\lim_{{x \to 0}} fg(x) = \lim_{{x \to 0}} f(x)g(x) = \lim_{{x \to 0}} f(x) \cdot \lim_{{x \to 0}} g(x) = \infty \cdot 0 = undefined\].

Compute the limits for (b)

For \(f(x) = -\frac{2}{x^3}\), as \(x\) approaches \(0\), \(f(x)\) approaches negative infinity: \[\lim_{{x \to 0}} f(x) = \lim_{{x \to 0}} -\frac{2}{x^3} = - \infty\]. For \(g(x) = 4x^3\), as \(x\) approaches \(0\), \(g(x)\) approaches \(0\): \[\lim_{{x \to 0}} g(x) = \lim_{{x \to 0}} 4x^3 = 0\]. To find the limit of the product \(fg\), apply the limit product rule: \[\lim_{{x \to 0}} fg(x) = \lim_{{x \to 0}} f(x)g(x) = \lim_{{x \to 0}} f(x) \cdot \lim_{{x \to 0}} g(x) = - \infty \cdot 0 = undefined\].