https://file.aiquickdraw.com/mathbot/file/17261207955918b2y1457.jpg

The expression in the image appears to represent a limit problem:

\[
\lim_{x \to 0} \frac{2x - \sin x}{| \sin x |}
\]

We are asked to compute the limit as \( x \) approaches 0. Let's proceed with solving it.

### Step 1: Analyze the behavior around \( x = 0 \)

We know that for small values of \( x \):
\[
\sin x \approx x
\]
This is based on the first term of the Taylor expansion of \( \sin x \) around \( x = 0 \).

### Step 2: Substitute and approximate
Near \( x = 0 \), we can approximate \( \sin x \approx x \), so the expression becomes:

\[
\lim_{x \to 0} \frac{2x - \sin x}{|\sin x|} \approx \lim_{x \to 0} \frac{2x - x}{|x|}
\]

This simplifies to:

\[
\lim_{x \to 0} \frac{x}{|x|}
\]

### Step 3: Evaluate the limit
Now we can evaluate the limit by considering the behavior of the absolute value function in the numerator and denominator.

- As \( x \to 0^+ \) (from the right), \( |x| = x \), so the expression becomes:
  \[
  \frac{x}{x} = 1
  \]
- As \( x \to 0^- \) (from the left), \( |x| = -x \), so the expression becomes:
  \[
  \frac{x}{-x} = -1
  \]

### Step 4: Conclusion
Since the limit approaches different values from the left and right, the two-sided limit does not exist. Therefore, the limit is **undefined**.

Would you like to ask more questions or get further details on this problem?

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Here are 8 related questions:

1. What is the Taylor series expansion of \( \sin x \) around \( x = 0 \)?
2. How do you calculate one-sided limits for piecewise functions?
3. What does it mean for a limit to be undefined?
4. How can absolute value functions affect the behavior of limits?
5. Can you use L'Hôpital's Rule to solve similar limit problems?
6. What is the relationship between the derivative of \( \sin x \) and the limit as \( x \to 0 \)?
7. How can you apply squeeze theorem in solving limit problems?
8. What other approximation techniques are useful for limits involving trigonometric functions?

**Tip:** When working with limits involving absolute value, always check one-sided limits, as the behavior might differ depending on the direction of approach.

What is the limit of (2x - sin(x)) / |sin(x)| as x approaches 0?

Explore how to calculate the limit of (2x - sin(x)) / |sin(x)| as x approaches 0. This problem involves trigonometric approximations, absolute value behavior, and evaluating one-sided limits, leading to the conclusion that the limit is undefined.

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