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To solve the given limit

\[ \lim_{x \to -7} \frac{(x + 7)^4 \sin\left(\frac{1}{(x + 7)^2}\right)}{\sin\left((x + 7)^2\right)}, \]

we can approach it by using some limit properties and approximations for small angles. Let's denote \( y = x + 7 \). As \( x \to -7 \), we have \( y \to 0 \). Thus, the limit becomes:

\[ \lim_{y \to 0} \frac{y^4 \sin\left(\frac{1}{y^2}\right)}{\sin(y^2)}. \]

For small \( y \), the approximations \(\sin(z) \approx z\) can be useful. However, \( \sin\left(\frac{1}{y^2}\right) \) oscillates rapidly and is bounded between -1 and 1. Therefore:

1. Consider the upper and lower bounds of \( \sin\left(\frac{1}{y^2}\right) \):

\[ -1 \leq \sin\left(\frac{1}{y^2}\right) \leq 1. \]

2. Now let's examine the effect of \( y^4 \sin\left(\frac{1}{y^2}\right) \) for \( y \to 0 \):

Since \( y^4 \) approaches 0 as \( y \to 0 \), \( y^4 \sin\left(\frac{1}{y^2}\right) \) also approaches 0, regardless of the oscillations of \(\sin\left(\frac{1}{y^2}\right)\).

3. For the denominator \(\sin(y^2)\):

As \( y \to 0 \),

\[ \sin(y^2) \approx y^2. \]

4. Substitute these approximations back into the limit:

\[ \lim_{y \to 0} \frac{y^4 \sin\left(\frac{1}{y^2}\right)}{\sin(y^2)} \approx \lim_{y \to 0} \frac{y^4 \sin\left(\frac{1}{y^2}\right)}{y^2} = \lim_{y \to 0} y^2 \sin\left(\frac{1}{y^2}\right). \]

5. Since \( y^2 \sin\left(\frac{1}{y^2}\right) \) is bounded by \( -y^2 \) and \( y^2 \):

\[ -y^2 \leq y^2 \sin\left(\frac{1}{y^2}\right) \leq y^2. \]

As \( y \to 0 \), \( y^2 \to 0 \). Thus, by the Squeeze Theorem:

\[ \lim_{y \to 0} y^2 \sin\left(\frac{1}{y^2}\right) = 0. \]

Therefore, the limit is:

\[ \lim_{x \to -7} \frac{(x + 7)^4 \sin\left(\frac{1}{(x + 7)^2}\right)}{\sin\left((x + 7)^2\right)} = 0. \]

Would you like any more details or have any questions about this solution?

Here are 5 related questions for further practice:

1. Evaluate \(\lim_{x \to 0} \frac{x^3 \sin\left(\frac{1}{x}\right)}{\sin(x)}\).
2. Find \(\lim_{x \to 0} \frac{\sin(3x)}{\sin(5x)}\).
3. Compute \(\lim_{x \to 0} \frac{x^2 \cos\left(\frac{1}{x}\right)}{\sin(x^2)}\).
4. Determine \(\lim_{x \to 0} \frac{\sin(x^3)}{x^3}\).
5. Solve \(\lim_{x \to \infty} \frac{\sin(x)}{x}\).

**Tip:** When dealing with trigonometric functions involving limits, it is often useful to employ small-angle approximations and the Squeeze Theorem to handle oscillatory behavior.

Evaluate the limit \( \lim_{x \to -7} \frac{(x + 7)^4}{\sin((x + 7)^2)} \sin \left( \frac{1}{(x + 7)^2} \right) \).

This problem evaluates the limit \( \lim_{x \to -7} \frac{(x + 7)^4}{\sin((x + 7)^2)} \sin \left( \frac{1}{(x + 7)^2} \right) \), using the Squeeze Theorem and small-angle approximations. The oscillating nature of the sine function and its approximation are key concepts in this solution.

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