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Question
\[\lim_{x \to \infty} \frac{\left| x \right|}{x}\] is equal to
Options
1
−1
0
does not exist
Solution
We know that,
\[\left| x \right| = \begin{cases}x, & if x \geq 0 \\ - x, & if x < 0\end{cases}\]
\[ \therefore \frac{\left| x \right|}{x} = \begin{cases}\frac{x}{x}, & if x \geq 0 \\ \frac{- x}{x}, & if x < 0\end{cases} = \begin{cases}1, & if x \geq 0 \\ - 1, & if x < 0\end{cases}\]
Now, for all x ≥ 0 (however, x may large be),
\[\frac{\left| x \right|}{x} = 1\]
\[\therefore \lim_{x \to \infty} \frac{\left| x \right|}{x} = 1\]
Hence, the correct answer is option (a).
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