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. \(\underset{x\to a}{\mathop{\lim }}\,\frac{{{\log }_{a}}\left( 1+x \right)}{x}={{\log }_{a}}e\)