Which derivative is described by the following expression?
$$
\lim _{x \rightarrow \pi} \frac{\sin (x)-0}{x-\pi}
$$

Choose 1 answer:
(A) $f^{\prime}(\pi)$, where $f(x)=\frac{\sin (x)}{x}$
(B) $f^{\prime}(0)$, where $f(x)=\sin (x)-x$
(C) $f^{\prime}(\pi)$, where $f(x)=\sin (x)$
(D) $f^{\prime}(0)$, where $f(x)=\sin (x)$

Question

Which derivative is described by the following expression?
$$
\lim _{x \rightarrow \pi} \frac{\sin (x)-0}{x-\pi}
$$

Choose 1 answer:
(A) $f^{\prime}(\pi)$, where $f(x)=\frac{\sin (x)}{x}$
(B) $f^{\prime}(0)$, where $f(x)=\sin (x)-x$
(C) $f^{\prime}(\pi)$, where $f(x)=\sin (x)$
(D) $f^{\prime}(0)$, where $f(x)=\sin (x)$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Consider the given limit expression and recognize it as the definition of the derivative of a function at a point. ‖ ‖ step 2 ⋮ The expression $\lim _{x \rightarrow \pi} \frac{\sin (x)-0}{x-\pi}$ is the definition of the derivative of $\sin(x)$ at $x = \pi$. ‖ ‖ step 3 ⋮ The derivative of $\sin(x)$ is $\cos(x)$, and evaluating this at $x = \pi$ gives $\cos(\pi) = -1$. ‖ ‖ step 4 ⋮ The limit expression simplifies to $-1$, which matches the derivative of $\sin(x)$ at $x = \pi$. ‖ ***C*** ∻Key Concept∻ ⚹Definition of the Derivative⚹ ∻Explanation∻ ⚹The limit expression given is the definition of the derivative of the function $\sin(x)$ at the point $x = \pi$.⚹◊What is the derivative of $f(x) = \frac{\sin (x)}{x}$ at $x = \pi$?⍭ Generate me a similar question◊

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