# Evaluation of Trigonometric, Exponential and Logarithm Limits

## Evaluation of Trigonometric, Exponential and Logarithm Limits

In order to evaluate trigonometric limits the following results are very useful:

1. $$\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x}{x}=1$$
2. $$\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan x}{x}=1$$
3. $$\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\sin }^{-1}}x}{x}=1$$
4. $$\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\tan }^{-1}}x}{x}=1$$
5. $$\underset{x\to a}{\mathop{\lim }}\,\frac{\sin \left( x-a \right)}{x-a}=1$$
6. $$\underset{x\to a}{\mathop{\lim }}\,\frac{\tan \left( x-a \right)}{x-a}=1$$
7. $$\underset{x\to 0}{\mathop{\lim }}\,\frac{\left| \sin x \right|}{x}$$does not exist
8. $$\underset{x\to 0}{\mathop{\lim }}\,\frac{\left| \tan x \right|}{x}$$ does not exist
9. $$\underset{x\to a}{\mathop{\lim }}\,\frac{\left| \sin \left( x-a \right) \right|}{x-a}$$ does not exist
10. $$\underset{x\to a}{\mathop{\lim }}\,\frac{\left| \tan \left( x-a \right) \right|}{x-a}$$ does not exist.

Expansions useful in Evaluation of Limits:

1. $${{\left( 1+x \right)}^{n}}=1+nx+\frac{n\left( n-1 \right)}{2!}{{x}^{2}}+\frac{n\left( n-1 \right)\left( n-2 \right)}{3!}{{x}^{3}}+…..$$
2. $${{e}^{x}}=1+\frac{x}{1!}+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}$$
3. $${{a}^{x}}=+\frac{x}{1!}\log _{e}^{a}+\frac{{{x}^{2}}}{2!}{{\left( \log _{e}^{a} \right)}^{2}}+……$$
4. $$\log \left( 1+x \right)=x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}-\frac{{{x}^{4}}}{4}+…..$$
5. $$\log \left( 1-x \right)=-x-\frac{{{x}^{2}}}{2}-\frac{{{x}^{3}}}{3}-\frac{{{x}^{4}}}{4}-…..$$
6. $$\sin x=x+\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}+…..$$
7. $$\cos x=1+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!}+………$$
8. $$\tan x=x+\frac{{{x}^{3}}}{3}+\frac{2}{15}{{x}^{5}}+……$$

Evaluate: $$\underset{x\to {{\tan }^{-1}}3}{\mathop{\lim }}\,\frac{{{\tan }^{2}}x-2\tan x-3}{{{\tan }^{2}}x-4\tan x+3}$$

Solution: we have,

$$\underset{x\to {{\tan }^{-1}}3}{\mathop{\lim }}\,\frac{{{\tan }^{2}}x-2\tan x-3}{{{\tan }^{2}}x-4\tan x+3}$$

$$=\underset{x\to {{\tan }^{-1}}3}{\mathop{\lim }}\,\frac{\left( \tan x-3 \right)\left( \tan x+1 \right)}{\left( \tan x-3 \right)\left( \tan x-1 \right)}$$

$$=-\underset{x\to {{\tan }^{-1}}3}{\mathop{\lim }}\,\frac{\tan x+1}{\tan x-1}=\frac{3+1}{3-1}=2$$

Evaluation of Exponential and Logarithm Limits:

1. $$\underset{x\to a}{\mathop{\lim }}\,\frac{{{a}^{x}}-1}{x}={{\log }_{e}}a,a>0$$
2. $$\underset{x\to a}{\mathop{\lim }}\,\frac{{{\log }_{e}}\left( 1+x \right)}{x}=1$$
3. $$\underset{x\to a}{\mathop{\lim }}\,\frac{{{e}^{x}}-1}{x}={{\log }_{e}}=1$$
4. $$\underset{x\to a}{\mathop{\lim }}\,\frac{{{\log }_{a}}\left( 1+x \right)}{x}={{\log }_{a}}e$$