Evaluation of Trigonometric, Exponential and Logarithm Limits
In order to evaluate trigonometric limits the following results are very useful:
- \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x}{x}=1\)
- \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan x}{x}=1\)
- \(\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\sin }^{-1}}x}{x}=1\)
- \(\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\tan }^{-1}}x}{x}=1\)
- \(\underset{x\to a}{\mathop{\lim }}\,\frac{\sin \left( x-a \right)}{x-a}=1\)
- \(\underset{x\to a}{\mathop{\lim }}\,\frac{\tan \left( x-a \right)}{x-a}=1\)
- \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\left| \sin x \right|}{x}\)does not exist
- \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\left| \tan x \right|}{x}\) does not exist
- \(\underset{x\to a}{\mathop{\lim }}\,\frac{\left| \sin \left( x-a \right) \right|}{x-a}\) does not exist
- \(\underset{x\to a}{\mathop{\lim }}\,\frac{\left| \tan \left( x-a \right) \right|}{x-a}\) does not exist.
Expansions useful in Evaluation of Limits:
- \({{\left( 1+x \right)}^{n}}=1+nx+\frac{n\left( n-1 \right)}{2!}{{x}^{2}}+\frac{n\left( n-1 \right)\left( n-2 \right)}{3!}{{x}^{3}}+…..\)
- \({{e}^{x}}=1+\frac{x}{1!}+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}\)
- \({{a}^{x}}=+\frac{x}{1!}\log _{e}^{a}+\frac{{{x}^{2}}}{2!}{{\left( \log _{e}^{a} \right)}^{2}}+……\)
- \(\log \left( 1+x \right)=x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}-\frac{{{x}^{4}}}{4}+…..\)
- \(\log \left( 1-x \right)=-x-\frac{{{x}^{2}}}{2}-\frac{{{x}^{3}}}{3}-\frac{{{x}^{4}}}{4}-…..\)
- \(\sin x=x+\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}+…..\)
- \(\cos x=1+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!}+………\)
- \(\tan x=x+\frac{{{x}^{3}}}{3}+\frac{2}{15}{{x}^{5}}+……\)
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:
- \(\underset{x\to a}{\mathop{\lim }}\,\frac{{{a}^{x}}-1}{x}={{\log }_{e}}a,a>0\)
- \(\underset{x\to a}{\mathop{\lim }}\,\frac{{{\log }_{e}}\left( 1+x \right)}{x}=1\)
- \(\underset{x\to a}{\mathop{\lim }}\,\frac{{{e}^{x}}-1}{x}={{\log }_{e}}=1\)
- \(\underset{x\to a}{\mathop{\lim }}\,\frac{{{\log }_{a}}\left( 1+x \right)}{x}={{\log }_{a}}e\)