Pregunta
Evaluate \( \lim _{x \rightarrow 3 \pi} \sin (x-3 \pi) \) Enter the exact answer. \( \lim _{x \rightarrow 3 \pi} \sin (x-3 \pi)= \) Number
Ask by Gordon Ball. in the United States
Mar 12,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The limit is \(0\).
Solución
1. The limit is given by
\[
\lim_{x \rightarrow 3\pi} \sin (x-3\pi)
\]
2. Notice that as \( x \rightarrow 3\pi \), the expression \( x-3\pi \) approaches 0:
\[
\lim_{x \rightarrow 3\pi} (x-3\pi) = 0
\]
3. Substitute this result into the sine function:
\[
\sin (x-3\pi) \rightarrow \sin(0)
\]
4. Since \(\sin(0) = 0\), we conclude that:
\[
\lim_{x \rightarrow 3\pi} \sin (x-3\pi) = 0
\]
5. Therefore, the exact answer is \(0\).
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Como
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Extra Insights
To evaluate the limit \( \lim _{x \rightarrow 3 \pi} \sin (x-3 \pi) \), we can start by substituting \( x \) with \( 3\pi \): \[ \sin(x - 3\pi) = \sin(3\pi - 3\pi) = \sin(0) \] Since \( \sin(0) = 0 \), we have: \[ \lim _{x \rightarrow 3 \pi} \sin (x-3 \pi) = 0 \] Thus, the exact answer is: \[ \lim _{x \rightarrow 3 \pi} \sin (x-3 \pi) = 0 \]
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