MHT CET · Maths · Limits
\(\lim _{x \rightarrow 0} \frac{\cos (m x)-\cos (n x)}{x^2}=\)
- A \(\frac{\mathrm{m}^2-\mathrm{n}^2}{2}\)
- B \(m^2-n^2\)
- C \(\frac{\mathrm{n}^2-\mathrm{m}^2}{2}\)
- D \(n^2-m^2\)
Answer & Solution
Correct Answer
(C) \(\frac{\mathrm{n}^2-\mathrm{m}^2}{2}\)
Step-by-step Solution
Detailed explanation
\(\lim _{x \rightarrow 0} \frac{\cos (m x)-\cos (n x)}{x^2} \)
\( =\lim _{x \rightarrow 0} \frac{\left[-2 \sin \frac{(m+n) x}{2} \sin \frac{(m-n) x}{2}\right]}{x^2} \)
\( =-2 \lim _{x \rightarrow 0}\left[\frac{\sin \left(\frac{m+n}{2}\right) x}{\left(\frac{m+n}{2}\right) x} \times\left(\frac{m+n}{2}\right)\right]\) \(\left[\frac{\sin \left(\frac{m-n}{2}\right) x}{\left(\frac{m-n}{2}\right) x} \times\left(\frac{m-n}{2}\right)\right] \)
\( =(-2)\left(\frac{m+n}{2}\right)\left(\frac{m-n}{2}\right)=\frac{m^2-n^2}{-2}=\frac{n^2-m^2}{2}\)
\( =\lim _{x \rightarrow 0} \frac{\left[-2 \sin \frac{(m+n) x}{2} \sin \frac{(m-n) x}{2}\right]}{x^2} \)
\( =-2 \lim _{x \rightarrow 0}\left[\frac{\sin \left(\frac{m+n}{2}\right) x}{\left(\frac{m+n}{2}\right) x} \times\left(\frac{m+n}{2}\right)\right]\) \(\left[\frac{\sin \left(\frac{m-n}{2}\right) x}{\left(\frac{m-n}{2}\right) x} \times\left(\frac{m-n}{2}\right)\right] \)
\( =(-2)\left(\frac{m+n}{2}\right)\left(\frac{m-n}{2}\right)=\frac{m^2-n^2}{-2}=\frac{n^2-m^2}{2}\)
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