Robbins King
02/14/2024 · Elementary School
Find the following limit. \[ \lim _{x \rightarrow 0^{+}}(x \csc (2 \pi-x)) \]
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Step-by-step Solution
To find the limit
\[
\lim_{x \rightarrow 0^{+}}(x \csc (2 \pi - x)),
\]
we start by rewriting the expression using the definition of cosecant:
\[
\csc(2\pi - x) = \frac{1}{\sin(2\pi - x)}.
\]
Using the property of the sine function, we have:
\[
\sin(2\pi - x) = \sin x.
\]
Thus, we can rewrite the limit as:
\[
\lim_{x \rightarrow 0^{+}}(x \csc(2\pi - x)) = \lim_{x \rightarrow 0^{+}} \left( x \cdot \frac{1}{\sin(2\pi - x)} \right) = \lim_{x \rightarrow 0^{+}} \left( \frac{x}{\sin x} \right).
\]
Next, we know from the standard limit that:
\[
\lim_{x \rightarrow 0} \frac{x}{\sin x} = 1.
\]
Therefore, we can conclude that:
\[
\lim_{x \rightarrow 0^{+}} \frac{x}{\sin x} = 1.
\]
Putting this together, we find:
\[
\lim_{x \rightarrow 0^{+}}(x \csc(2\pi - x)) = 1.
\]
Thus, the final answer is:
\[
\boxed{1}.
\]
Quick Answer
The limit is 1.
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