https://file.aiquickdraw.com/mathbot/file/1725502023956qecv1k9h.jpg

The provided image contains the following mathematical expression:

\[
\lim_{{x \to 0}} \frac{\sin^2(4x)}{x^2(1 - \cos(5x))}
\]

The possible answer choices are:

- A: 2
- B: 1
- C: 0
- D: \(-\frac{1}{5}\)
- E: -1

Let's evaluate this limit step by step:

### Step 1: Use known approximations for small \(x\)
As \(x \to 0\), we can use the small-angle approximations:
- \(\sin(x) \approx x\)
- \(\cos(x) \approx 1 - \frac{x^2}{2}\)

### Step 2: Approximate the numerator
\[
\sin^2(4x) \approx (4x)^2 = 16x^2
\]

### Step 3: Approximate the denominator
\[
1 - \cos(5x) \approx 1 - \left(1 - \frac{(5x)^2}{2}\right) = \frac{25x^2}{2}
\]

Thus, the expression becomes approximately:

\[
\frac{16x^2}{x^2 \cdot \frac{25x^2}{2}} = \frac{16x^2}{\frac{25x^2}{2}} = \frac{16 \cdot 2}{25} = \frac{32}{25}
\]

### Step 4: Conclusion
The limit of the expression as \(x \to 0\) is \(\frac{32}{25}\), which is close to 1.28. However, none of the provided options exactly match this result.

It seems like there is a mismatch between the approximations used and the answer choices. Let's check if any approximations in the context of a multiple-choice exam match the choices better.

The closest approximation in the options provided is **B: 1**.

Would you like more details on this, or do you have any questions?

### Related Questions:
1. How do small-angle approximations help simplify trigonometric limits?
2. What other techniques can be used to evaluate limits involving trigonometric functions?
3. How does L'Hopital's Rule apply to limits of indeterminate forms?
4. What is the behavior of the sine and cosine functions as \(x \to 0\)?
5. How do Taylor series expansions assist in limit evaluations?

**Tip:** When working with trigonometric limits, approximations such as \(\sin(x) \approx x\) and \(\cos(x) \approx 1 - \frac{x^2}{2}\) for small \(x\) are extremely useful for simplifying complex expressions.

Evaluate the limit: lim_(x -> 0) (sin^2(4x) / (x^2(1 - cos(5x))) with the options: A) 2, B) 1, C) 0, D) -1/5, E) -1.

This problem explores the limit of the expression (sin^2(4x)) / (x^2(1 - cos(5x))) as x approaches 0, using trigonometric approximations for small x values. It is an example of applying small angle approximation to simplify and evaluate limits.

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