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Question
\[\lim_{x \to 1} \frac{1 - \frac{1}{x}}{\sin \pi \left( x - 1 \right)}\]
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Solution
\[\lim_{x \to 1} \left[ \frac{1 - \frac{1}{x}}{sin\pi\left( x - 1 \right)} \right]\]
\[ = \lim_{x \to 1} \left[ \frac{x - 1}{xsin\pi\left( x - 1 \right)} \right]\]
Let y = x – 1
If x → 1, then y → 0.
\[= \lim_{y \to 0} \left[ \frac{y}{\left( y + 1 \right)sin\pi y} \right]\]
\[ = \lim_{y \to 0} \left[ \frac{1}{\pi\left( y + 1 \right) \frac{\sin \pi y}{\pi y}} \right]\]
\[ = \frac{1}{\pi\left( 0 + 1 \right) \times 1}\]
\[ = \frac{1}{\pi}\]
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