Question
Download Solution PDF\(\rm \displaystyle \lim_{\theta\rightarrow 0}\frac{1-\cos m \theta}{1-\cos n\theta}=?\)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
\(\rm \displaystyle \lim_{θ\rightarrow 0}\frac{1-\cos m θ}{1-\cos nθ}\)
Concept:
Use concept of limit .
\(\rm \lim_{x\rightarrow0}\frac{sinx}{x}=1\)
And formula \(\rm cos2θ=1-2sin^2θ\)
Calculation:
\(\rm \displaystyle \lim_{θ\rightarrow 0}\frac{1-\cos m θ}{1-\cos nθ}\)
\(\rm \displaystyle =\lim_{θ\rightarrow 0}\frac{1-(1-2\sin^2 \frac{m θ}{2})}{1-(1-2\sin^2 \frac{nθ}{2})}\)
\(\rm \displaystyle =\lim_{θ\rightarrow 0}\frac{2\sin^2 \frac{m θ}{2}}{2\sin^2 \frac{nθ}{2}}\)
\(\rm \displaystyle =\frac{\lim_{θ\rightarrow 0}\sin^2 \frac{m θ}{2}}{\lim_{θ\rightarrow 0}\sin^2 \frac{nθ}{2}}\)
multiply and divide by m2θ/n2θ
\(\rm =\frac{m^2θ}{n^2θ}=\frac{m^2}{n^2}\)
Hence the option (4) is correct.
Last updated on May 26, 2025
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