Evaluate the following limit: _ x1 1-1 x (x-1) - Vidyadip

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{1}}\frac{1-\frac{1}{\text{x}}}{\sin\pi(\text{x}-1)}$

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Answer

$\lim\limits_{\text{x}\rightarrow{1}}\frac{1-\frac{1}{\text{x}}}{\sin\pi(\text{x}-1)}$ As x → 1, then x - 1 → 0 let x - 1 = y $=\lim\limits_{{\text{x}-{1\rightarrow0}}}\frac{(\text{x}-1)}{\text{x}\times\sin\pi(\text{x}-1)}$ $=\lim\limits_{{\text{y}\rightarrow0}}\frac{\text{y}}{(\text{y}+1)\sin(\pi\text{y})}$ $=\lim\limits_{{\text{y}\rightarrow0}}\frac{\text{y}}{(\text{y}+1)\sin(\pi\text{y})}$ $=\lim\limits_{{\text{y}\rightarrow0}}\frac{1}{\frac{(\text{y}+1)\sin(\pi\text{y})}{\text{y}}}$ $=\frac{1}{\Big(\lim\limits_{\text{y}\rightarrow0}(\text{y}+1)\Big)\times\Big(\lim\limits_{\text{y}\rightarrow0}\frac{\sin\pi\text{y}}{\text{y}\times\pi}\times\pi\Big)}$ $=\frac{1}{(1)(1\times\pi)}$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1\Big]$ $=\frac{1}{\pi}$

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