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We are tasked with finding the following limit:

\[
\lim_{{x \to 7}} \frac{{\sin(x - 7)}}{{x^2 + 2x - 63}}
\]

### Step 1: Substituting \( x = 7 \) directly

Let's start by substituting \( x = 7 \) into the expression:

- The numerator becomes:
  \[
  \sin(7 - 7) = \sin(0) = 0
  \]
  
- The denominator becomes:
  \[
  7^2 + 2(7) - 63 = 49 + 14 - 63 = 0
  \]

This gives a \(\frac{0}{0}\) indeterminate form, so we need to apply **L'Hopital's Rule**, which is applicable in cases of indeterminate forms like \( \frac{0}{0} \).

### Step 2: Apply L'Hopital's Rule

L'Hopital's Rule states that if the limit results in an indeterminate form, we can differentiate the numerator and the denominator separately and then take the limit again.

1. **Differentiate the numerator**: 
   \[
   \frac{d}{dx} [\sin(x - 7)] = \cos(x - 7)
   \]

2. **Differentiate the denominator**:
   \[
   \frac{d}{dx} [x^2 + 2x - 63] = 2x + 2
   \]

Now, the limit becomes:
\[
\lim_{{x \to 7}} \frac{{\cos(x - 7)}}{{2x + 2}}
\]

### Step 3: Substituting \( x = 7 \) into the new expression

- The numerator becomes:
  \[
  \cos(7 - 7) = \cos(0) = 1
  \]

- The denominator becomes:
  \[
  2(7) + 2 = 14 + 2 = 16
  \]

Thus, the limit is:
\[
\lim_{{x \to 7}} \frac{{1}}{{16}} = \frac{1}{16}
\]

### Final Answer:
\[
\boxed{\frac{1}{16}}
\]

Would you like further details on any step or more explanations?

### Related Questions:
1. What are the general conditions for applying L'Hopital's Rule?
2. How do you differentiate trigonometric functions like \(\sin(x)\) and \(\cos(x)\)?
3. Why does \(\sin(0) = 0\) and \(\cos(0) = 1\)?
4. What other methods could solve limits involving trigonometric functions?
5. How does factoring help in solving limits when L'Hopital's Rule doesn't apply?

#### Tip:
Always check for indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) before applying L'Hopital’s Rule.

Find the limit: lim(x→7) (sin(x - 7)) / (x^2 + 2x - 63)

Learn how to calculate the limit lim(x→7) (sin(x - 7)) / (x^2 + 2x - 63) using L'Hopital's Rule. This problem involves differentiating both the numerator and denominator, and applying trigonometric identities to solve.

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