# Evolution of Exponential and Logarithmic Limits

## 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: $$\underset{x\to 0}{\mathop{\lim }}\,\frac{{{2}^{x}}-1}{\sqrt{1+x}-1}$$.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{{{2}^{x}}-1}{\sqrt{1+x}-1}\times \frac{\sqrt{1+x}+1}{\sqrt{1+x}+1}$$.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{({{2}^{x}}-1)\times \sqrt{1+x}+1}{1+x-1}$$.

$$\underset{x\to 0}{\mathop{\lim }}\,\frac{({{2}^{x}}-1)}{x}\times \underset{x\to 0}{\mathop{\lim }}\,\sqrt{1+x}+1$$.

$$(\log 2)2=\log 4$$.