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Question
\[\lim_{x \to 3} \frac{x - 3}{\left| x - 3 \right|},\] is equal to
Options
1
−1
0
does not exist
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Solution
does not exist
\[\lim_{x \to 3} \frac{x - 3}{\left| x - 3 \right|}\]
\[\text{ LHL at } x = 3: \]
\[ \lim_{x \to 3^-} \frac{x - 3}{- \left( x - 3 \right)} \left[ \because \left| x - 3 \right| = - \left( x - 3 \right), \text{ when } x < 3 \right]\]
\[ \Rightarrow - 1\]
\[\text{ RHL at } x = 3: \]
\[ \lim_{x \to 3^+} \frac{x - 3}{x - 3} \left[ \because \left| x - 3 \right| = x - 3, \text{ when } x > 3 \right]\]
\[ = 1\]
LHL ≠ RHL
Therefore, limit does not exist.
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