Limits of the Form \(\underset{x\to a}{\mathop{\lim }}\,{{\left( f(x) \right)}^{g(x)}}\)–\({{1}^{\infty }}\)
1. \(\underset{x\to 0}{\mathop{\lim }}\,{{\left( 1+x \right)}^{\frac{1}{x}}}=e\ \ (or)\ \ \ \ \ \underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{1}{x} \right)}^{x}}=e\)
\(=\underset{x\to 0}{\mathop{\lim }}\,\left( 1+\frac{1}{x}x+\frac{\frac{1}{x}\left( \frac{1}{x}-1 \right)}{2!}{{x}^{2}}+\frac{\frac{1}{x}\left( \frac{1}{x}-1 \right)\left( \frac{1}{x}-2 \right)}{2!}{{x}^{3}}+…. \right)\)\(=\underset{x\to 0}{\mathop{\lim }}\,\left( 1+1+\frac{1(1-x)}{2!}+\frac{1(1-x)(1-2x)}{3!}+… \right)\),
\(=\left( 1+1+\frac{1}{2!}+\frac{1}{3!}+…. \right)=e\)2. \(L=\underset{x\to a}{\mathop{\lim }}\,f{{\left( x \right)}^{g(x)}}\). If \(\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=1\ \ and\ \ \underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=\infty \)
Then \(L=\underset{x\to a}{\mathop{\lim }}\,f{{\left( x \right)}^{g(x)}}\),
\(L=\underset{x\to a}{\mathop{\lim }}\,{{\left\{ 1+\left( f\left( x \right)-1 \right) \right\}}^{\frac{1}{f\left( x \right)-1}\left( f\left( x \right)-1 \right)\times g\left( x \right)}}\),
\(L=\underset{x\to a}{\mathop{\lim }}\,{{\left\{ {{\left( 1+\left( f\left( x \right)-1 \right) \right)}^{\frac{1}{f\left( x \right)-1}}} \right\}}^{\underset{x\to a}{\mathop{\lim }}\,}}^{\left( f\left( x \right)-1 \right)\times g\left( x \right)}\),
\(={{e}^{\underset{x\to a}{\mathop{\lim }}\,\left( f\left( x \right)-1 \right)\times g\left( x \right)}}\).
Example: Evaluate \(\underset{x\to 0}{\mathop{\lim }}\,{{\left( 1+x \right)}^{\cos ecx}}\)
Solution:
\(\underset{x\to 0}{\mathop{\lim }}\,{{\left( 1+x \right)}^{\cos ecx}}\),
\(\underset{x\to 0}{\mathop{\lim }}\,{{\left( 1+x \right)}^{\frac{1}{\sin x}}}\),
\(\underset{x\to 0}{\mathop{\lim }}\,{{\left( 1+x \right)}^{\frac{1}{\sin x}\times \frac{x}{x}}}\),
\(\underset{x\to 0}{\mathop{\lim }}\,{{\left\{ \left( 1+x \right) \right\}}^{\frac{x}{\sin x}\times \frac{1}{x}}}\),
\(\underset{x\to 0}{\mathop{\lim }}\,{{\left\{ {{\left( 1+x \right)}^{\frac{1}{x}}} \right\}}^{\frac{x}{\sin x}}}\),
\(\underset{x\to 0}{\mathop{\lim }}\,{{\left\{ {{\left( 1+x \right)}^{\frac{1}{x}}} \right\}}^{\underset{x\to 0}{\mathop{\lim }}\,\ \ \frac{x}{\sin x}}}\),
\(\because \underset{x\to 0}{\mathop{\lim }}\,{{\left( 1+x \right)}^{\frac{1}{x}}}=e\),
\({{e}^{\underset{x\to 0}{\mathop{\lim }}\,\ \ \frac{x}{\sin x}}}\),
\(={{e}^{1}}\).