Limits of the Form \(\underset{x\to a}{\mathop{\lim }}\,{{\left( f(x) \right)}^{g(x)}}\) – Part 1
Limits of the Form \(\underset{x\to a}{\mathop{\lim }}\,{{\left( f(x) \right)}^{g(x)}}\)– Form: \({{0}^{0}},{{\infty }^{0}}\):
Let \(L\text{ }=\underset{x\to a}{\mathop{\lim }}\,{{\left( f(x) \right)}^{g(x)}}\) then
Taking log on both sides
\({{\log }_{e}}L\text{ }={{\log }_{e}}\left[ \underset{x\to a}{\mathop{\lim }}\,{{\left( f(x) \right)}^{g(x)}} \right]\),
\({{\log }_{e}}L\text{ }=\left[ {{\log }_{e}}\underset{x\to a}{\mathop{\lim }}\,{{\left( f(x) \right)}^{g(x)}} \right]\),
\({{\log }_{e}}L\text{ }=\left[ \underset{x\to a}{\mathop{\lim }}\,{{\log }_{e}}{{\left( f(x) \right)}^{g(x)}} \right]\),
\({{\log }_{e}}L\text{ }=\left[ \underset{x\to a}{\mathop{\lim }}\,g(x){{\log }_{e}}\left( f(x) \right) \right]\),
\(L\text{ }={{e}^{\left[ \underset{x\to a}{\mathop{\lim }}\,g(x){{\log }_{e}}\left( f(x) \right) \right]}}\).
Example: Evaluate \(\underset{x\to \infty }{\mathop{\lim }}\,{{x}^{1/x}}\)
Solution:
Given that \(\underset{x\to \infty }{\mathop{\lim }}\,{{x}^{1/x}}\)
Let us consider\(L=\underset{x\to \infty }{\mathop{\lim }}\,{{x}^{1/x}}\)
Taking log on both sides
\({{\log }_{e}}L={{\log }_{e}}\left( \underset{x\to \infty }{\mathop{\lim }}\,{{x}^{1/x}} \right)\),
\({{\log }_{e}}L=\left( \underset{x\to \infty }{\mathop{\lim }}\,{{\log }_{e}}{{x}^{1/x}} \right)\),
\({{\log }_{e}}L=\left( \underset{x\to \infty }{\mathop{\lim }}\,\ \frac{1}{x}\times {{\log }_{e}}x \right)\),
\(L={{e}^{\left( \underset{x\to \infty }{\mathop{\lim }}\,\ \frac{1}{x}\times {{\log }_{e}}x \right)}}\),
\(L={{e}^{0}}=1\).