Question
f) \( \lim _{x \rightarrow 0}(x \cdot \csc x) \)
Ask by Wilson Parsons. in the United States
Mar 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The limit is 1.
Solution
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The Deep Dive
To find \( \lim _{x \rightarrow 0}(x \cdot \csc x) \), we first rewrite \( \csc x \) as \( \frac{1}{\sin x} \). Thus, the limit becomes: \[ \lim_{x \rightarrow 0} (x \cdot \csc x) = \lim_{x \rightarrow 0} \left( x \cdot \frac{1}{\sin x} \right) = \lim_{x \rightarrow 0} \frac{x}{\sin x} \] We know from a standard limit that: \[ \lim_{x \rightarrow 0} \frac{x}{\sin x} = 1 \] Therefore, we have: \[ \lim_{x \rightarrow 0}(x \cdot \csc x) = 1 \] So, the final result is: \[ \boxed{1} \]
