KCET · Maths · Limits
If \(f(x)=\left|\begin{array}{ccc}\cos x & 1 & 0 \\ 0 & 2 \cos x & 3 \\ 0 & 1 & 2 \cos x\end{array}\right|\), then \(\lim _{x \rightarrow \pi} f(x)\) is equal to
- A \(-1\)
- B 1
- C 0
- D 3
Answer & Solution
Correct Answer
(A) \(-1\)
Step-by-step Solution
Detailed explanation
If \(f(x)=\left|\begin{array}{ccc}\cos x & 1 & 0 \\ 0 & 2 \cos x & 3 \\ 0 & 1 & 2 \cos x\end{array}\right|\)
Expand along \(C_{1}\),
\(\begin{aligned}
&=\cos x\left(4 \cos ^{2} x-3\right) \\
&=4 \cos ^{3} x-3 \cos x \\
&=\cos 3 x \\
\therefore \lim _{x \rightarrow \pi} \cos 3 x=\cos 3 \pi=-1
\end{aligned}\)
Expand along \(C_{1}\),
\(\begin{aligned}
&=\cos x\left(4 \cos ^{2} x-3\right) \\
&=4 \cos ^{3} x-3 \cos x \\
&=\cos 3 x \\
\therefore \lim _{x \rightarrow \pi} \cos 3 x=\cos 3 \pi=-1
\end{aligned}\)
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