Question:
Evaluate the Given limit: \(\lim_{x\rightarrow 0}(cosec\,x-cot\,x)\)
Evaluate the Given limit: \(\lim_{x\rightarrow 0}(cosec\,x-cot\,x)\)
Updated On: Oct 23, 2023
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Solution and Explanation
\(\lim_{x\rightarrow 0}(cosec\,x-cot\,x)\)
At x = 0, the value of the given function takes the form ∞ to -∞.
Now,
=\(\lim_{x\rightarrow 0}(cosec\,x-cot\,x)\)
= \(\lim_{x\rightarrow 0}(\frac{1}{x}-\frac{cos\,x}{sin\,x})\)
= \(\lim_{x\rightarrow 0}({1}-\frac{cos\,x}{sin\,x})\)
= \(\lim_{x\rightarrow 0}\) \(\frac{1-\frac{cis\,x}{x}}{\frac{x}{\frac{son\,x}{x}}}\)
=\(\frac{\lim_{x\rightarrow 0}1-\frac{cos\,x}{x}}{\lim_{x\rightarrow 0}\frac{sin\,x}{x}}\)
= \(\frac{0}{1}\) [= \(\lim_{x\rightarrow 0}\) ( \(1-\frac{cos\,x}{x}\)) = 0 and \(\lim_{x\rightarrow 0}\) \(\frac{sin\,x}{x}\) = 1]
=0
At x = 0, the value of the given function takes the form ∞ to -∞.
Now,
=\(\lim_{x\rightarrow 0}(cosec\,x-cot\,x)\)
= \(\lim_{x\rightarrow 0}(\frac{1}{x}-\frac{cos\,x}{sin\,x})\)
= \(\lim_{x\rightarrow 0}({1}-\frac{cos\,x}{sin\,x})\)
= \(\lim_{x\rightarrow 0}\) \(\frac{1-\frac{cis\,x}{x}}{\frac{x}{\frac{son\,x}{x}}}\)
=\(\frac{\lim_{x\rightarrow 0}1-\frac{cos\,x}{x}}{\lim_{x\rightarrow 0}\frac{sin\,x}{x}}\)
= \(\frac{0}{1}\) [= \(\lim_{x\rightarrow 0}\) ( \(1-\frac{cos\,x}{x}\)) = 0 and \(\lim_{x\rightarrow 0}\) \(\frac{sin\,x}{x}\) = 1]
=0
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