Question:
\[
\lim_{x\to0}
\frac{|x|\log_e(1+|\sin2x|)}
{x^2(|x|+3)}
\]
\[
\lim_{x\to0}
\frac{|x|\log_e(1+|\sin2x|)}
{x^2(|x|+3)}
\]
Show Hint
Near zero:
\[
\sin x\sim x,
\qquad
\log(1+x)\sim x.
\]
These approximations simplify many limit problems instantly.
Updated On: Jun 11, 2026
- Does not exist
- Exists and equals \(\frac23\)
- Exists and equals \(\frac13\)
- Exists and equals \(0\)
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Use standard approximations near \(x=0\).
\[ \sin2x\sim2x. \] Hence \[ |\sin2x| \sim2|x|. \] Also, \[ \log(1+t)\sim t \quad (t\to0). \] Therefore, \[ \log(1+|\sin2x|) \sim2|x|. \]
Step 2: Substitute into the limit.
\[ \frac{|x|\log(1+|\sin2x|)} {x^2(|x|+3)} \sim \frac{|x|(2|x|)} {x^2(|x|+3)} \] \[ = \frac{2x^2} {x^2(|x|+3)} \] \[ = \frac{2}{|x|+3}. \]
Step 3: Evaluate the limit.
\[ \lim_{x\to0} \frac{2}{|x|+3} = \frac23. \] Hence \[ \boxed{\frac23}. \]
\[ \sin2x\sim2x. \] Hence \[ |\sin2x| \sim2|x|. \] Also, \[ \log(1+t)\sim t \quad (t\to0). \] Therefore, \[ \log(1+|\sin2x|) \sim2|x|. \]
Step 2: Substitute into the limit.
\[ \frac{|x|\log(1+|\sin2x|)} {x^2(|x|+3)} \sim \frac{|x|(2|x|)} {x^2(|x|+3)} \] \[ = \frac{2x^2} {x^2(|x|+3)} \] \[ = \frac{2}{|x|+3}. \]
Step 3: Evaluate the limit.
\[ \lim_{x\to0} \frac{2}{|x|+3} = \frac23. \] Hence \[ \boxed{\frac23}. \]
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