Given lim_(x rarr7)f(x)=-2 and lim_(x rarr7)g(x)=4 , evaluate: lim_(x rarr7)(g(x)ln(-4f(x)+8))= lim_(x rarr7)(-4g(x)e^(8f(x)))= lim_(x rarr7)(sqrt(-4f(x)+8g(x)))=

<p></p><br /><p>1. To find \(\lim _{x \rightarrow 7}(g(x) \ln (-4 f(x)+8))\), we use the given limits \(\lim_{x \rightarrow 7} f(x) = -2\) and \(\lim_{x \rightarrow 7} g(x) = 4\). The limit of \(-4 f(x) + 8\) as \(x \rightarrow 7\) is \(8 - 4(-2) = 8 + 8 = 16\). Therefore, the limit becomes \(4 \ln(16) = 4 \ln(8) = 4 \ln(8)\).</p><br /><br /><p>2. To find \(\lim _{x \rightarrow 7}\left(-4 g(x) e^{8 f(x)}\right)\), we use the given limits \(\lim_{x \rightarrow 7} f(x) = -2\) and \(\lim_{x \rightarrow 7} g(x) = 4\). The limit becomes \(-4 \times 4 \times e^{8 \times -2} = -4 \times 4 \times e^{-16}\).</p><br /><br /><p>3. To find \(\lim _{x \rightarrow 7}(\sqrt{-4 f(x)+8 g(x)})\), we use the given limits \(\lim_{x \rightarrow 7} f(x) = -2\) and \(\lim_{x \rightarrow 7} g(x) = 4\). The limit of \(-4 f(x) + 8 g(x)\) as \(x \rightarrow 7\) is \(8 \times 4 - 4 \times -2 = 32 + 8 = 40\). Therefore, the limit becomes \(\sqrt{40} = 2 \sqrt{10} = 2 \sqrt{30}\).</p>