Evolution of Exponential and Logarithmic Limits
1. Evaluate type of limit \(\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}-1}{x}={{\log }_{e}}a\).
Proof: \(\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}-1}{x}\).
\(=\underset{x\to 0}{\mathop{\lim }}\,\frac{\left( 1+\frac{x(\log a)}{1!}+\frac{{{x}^{2}}{{(\log a)}^{2}}}{2!}+…… \right)-1}{x}\).
\(=\underset{x\to 0}{\mathop{\lim }}\,\frac{\left( \frac{x(\log a)}{1!}+\frac{{{x}^{2}}{{(\log a)}^{2}}}{2!}+…… \right)}{x}\).
\(={{\log }_{e}}a\).
2. Evaluate type of limit \(\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{x}}-1}{x}=1\).
Proof: \(\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{x}}-1}{x}\).
\(=\underset{x\to 0}{\mathop{\lim }}\,\frac{\left( 1+\frac{x(\log e)}{1!}+\frac{{{x}^{2}}{{(\log e)}^{2}}}{2!}+…… \right)-1}{x}\).
\(=\underset{x\to 0}{\mathop{\lim }}\,\frac{\left( \frac{x(\log e)}{1!}+\frac{{{x}^{2}}{{(\log e)}^{2}}}{2!}+…… \right)}{x}\).
\(=\underset{x\to 0}{\mathop{\lim }}\,\frac{\left( \frac{x}{1!}+\frac{{{x}^{2}}}{2!}+….. \right)}{x}\) = 1.
3. Evaluate type of limit \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\log (1+x)}{x}=1\).
Proof: \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\log (1+x)}{x}\).
\(\underset{x\to 0}{\mathop{\lim }}\,\frac{\left[ x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}-…….. \right]}{x}\).
\(\underset{x\to 0}{\mathop{\lim }}\,\left[ 1-\frac{x}{2}+\frac{{{x}^{2}}}{3}-…….. \right]\) = 1 .
Example 1: Evaluate \(\underset{x\to 0}{\mathop{\lim }}\,\frac{{{2}^{x}}-1}{\sqrt{1+x}-1}\).
Solution: \(\underset{x\to 0}{\mathop{\lim }}\,\frac{{{2}^{x}}-1}{\sqrt{1+x}-1}\).
\(\underset{x\to 0}{\mathop{\lim }}\,\frac{{{2}^{x}}-1}{\sqrt{1+x}-1}\times \frac{\sqrt{1+x}+1}{\sqrt{1+x}+1}\).
\(\underset{x\to 0}{\mathop{\lim }}\,\frac{({{2}^{x}}-1)\times \sqrt{1+x}+1}{1+x-1}\).
\(\underset{x\to 0}{\mathop{\lim }}\,\frac{({{2}^{x}}-1)}{x}\times \underset{x\to 0}{\mathop{\lim }}\,\sqrt{1+x}+1\).
\((\log 2)2=\log 4\).