KCET · Maths · Limits
If \(\lim _{x \rightarrow 0} \frac{\sin (2+x)-\sin (2-x)}{x}=A \cos B\), then the values of \(A\) and \(B\) respectively are
- A 1,2
- B 2,1
- C 1,1
- D 2,2
Answer & Solution
Correct Answer
(D) 2,2
Step-by-step Solution
Detailed explanation
We have,
\(\lim _{x \rightarrow 0} \frac{\sin (2+x)-\sin (2-x)}{x}=A \cos B\)
\(\lim _{x \rightarrow 0} \frac{2 \cos \left(\frac{2+x+2-x}{2}\right) \sin \left(\frac{2+x-2+x}{2}\right)}{x}\)
\(\begin{aligned} & =\lim _{x \rightarrow 0} 2 \cos 2 \frac{\sin x}{x} \\ & =2 \cos 2 \lim _{x \rightarrow 0} \frac{\sin x}{x}=2 \cos 2\end{aligned}\)
Thus, \(A=2\) and \(B=2\)
\(\lim _{x \rightarrow 0} \frac{\sin (2+x)-\sin (2-x)}{x}=A \cos B\)
\(\lim _{x \rightarrow 0} \frac{2 \cos \left(\frac{2+x+2-x}{2}\right) \sin \left(\frac{2+x-2+x}{2}\right)}{x}\)
\(\begin{aligned} & =\lim _{x \rightarrow 0} 2 \cos 2 \frac{\sin x}{x} \\ & =2 \cos 2 \lim _{x \rightarrow 0} \frac{\sin x}{x}=2 \cos 2\end{aligned}\)
Thus, \(A=2\) and \(B=2\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(A=\left[\begin{array}{ccc}0 & x & 16 \\ x & 5 & 7 \\ 0 & 9 & x\end{array}\right]\) is singular, then the possible values of \(x\) areKCET 2007 Easy
- If \(u=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)\) and \(v=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)\) then \(\frac{d u}{d v}\) isKCET 2023 Medium
- Sum of the squares of the order and degree (if defined) of a differential equation \(2 y^{\prime}+\left(y^{\prime \prime}\right)^2=\sqrt{y^{\prime \prime}-3}\) isKCET 2026 Medium
- The function \(f(x)=\sqrt{3} \sin 2 x-\cos 2 x+4\) is one-one in the intervalKCET 2021 Easy
- The value of \( \lim _{\theta \rightarrow 0} \frac{1-\cos 4 \theta}{1-\cos 6 \theta} \) isKCET 2017 Easy
- If \( A \) and \( B \) are finite sets and \( A \subset B \), thenKCET 2017 Easy
More PYQs from KCET
- A proton and an alpha particle are accelerated under the same potential difference. The ratio of de-Broglie wavelengths of the proton and the alpha particle isKCET 2012 Medium
- In a regular graph of 15 vertices the sum of the degree of the vertices is 60 . Then, the degree of each vertex isKCET 2013 Easy
- Adsorption theory is applicable forKCET 2013 Easy
- \(\int \frac{\cos ^{n-1} x}{\sin ^{n+1} x} d x(\) where, \(n \neq 0)\) is equal toKCET 2013 Medium
- The dc common emitter current gain of a n-p-n transistor is 50, The potential difference applied
across the collector and emitter of a transistor used in \( C E \) configuration is \( V_{C E}=2 V \). If the
collector resistance,
\( R_{C}=4 \mathrm{~K} \Omega \), the base current \( \left(I_{B}\right) \) and the collector current \( \left(I_{C}\right) \) areKCET 2018 Hard - A \(2 \mathrm{~g}\) object, located in a region of uniform electric field \(\mathbf{E}=\left(300 \mathrm{NC}^{-1}\right) \hat{\mathbf{i}}\) carries a charge \(Q\). The object released from rest at \(x=0\), has a kinetic energy of \(0.12 \mathrm{~J}\) at \(x=0.5 \mathrm{~m}\). Then, \(Q\) isKCET 2021 Medium