Show that _ x 0 1 x does not exist. - Vidyadip

Question

Show that $\lim\limits_{\text{x}\rightarrow0}\ \sin\frac{1}{\text{x}}$ does not exist.

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Answer

$\lim\limits_{\text{x}\rightarrow0^-}\sin\frac{1}{\text{x}}=\lim\limits_{\text{h}\rightarrow0}\sin\frac{1}{0-\text{h}}=-\lim\limits_{\text{h}\rightarrow0}\ \sin\frac{1}{\text{h}}$ = - (Anoscillating number which ascillates between - 1 and 1) So, $\lim\limits_{\text{x}\rightarrow0^-}\ \sin\frac{1}{\text{x}}$ does not exist. Similarly, $\lim\limits_{\text{x}\rightarrow0^+}\ \sin\frac{1}{\text{x}}$ does not exist $\lim\limits_{\text{x}\rightarrow0}\ \sin\frac{1}{\text{x}}$ does not exist.

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