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We are given the limit:

\[
\lim_{{y \to \pi}} \frac{{\sin^2 y}}{{1 + \cos y}}
\]

### Step-by-Step Solution:

1. **Evaluate the expression at \( y = \pi \):**

   - \( \sin(\pi) = 0 \)
   - \( \cos(\pi) = -1 \)

   Substituting these values into the expression:

   \[
   \frac{{\sin^2(\pi)}}{{1 + \cos(\pi)}} = \frac{{0^2}}{{1 + (-1)}} = \frac{0}{0}
   \]

   This gives an indeterminate form \( \frac{0}{0} \), which means we can apply **L'Hôpital's Rule**.

2. **Apply L'Hôpital's Rule:**

   L'Hôpital's Rule states that if we have an indeterminate form like \( \frac{0}{0} \), we can take the derivative of the numerator and denominator separately and then take the limit.

   - Derivative of \( \sin^2 y \) using the chain rule:
     \[
     \frac{d}{dy} \left( \sin^2 y \right) = 2\sin y \cdot \cos y
     \]
   - Derivative of \( 1 + \cos y \):
     \[
     \frac{d}{dy} \left( 1 + \cos y \right) = -\sin y
     \]

   Now the limit becomes:
   \[
   \lim_{{y \to \pi}} \frac{{2\sin y \cos y}}{{-\sin y}}
   \]

3. **Simplify the expression:**

   We can cancel out \( \sin y \) in the numerator and denominator (as \( y \to \pi \), \( \sin y \neq 0 \)):

   \[
   \lim_{{y \to \pi}} \frac{{2 \cos y}}{{-1}} = -2 \cos(\pi)
   \]

4. **Substitute \( \cos(\pi) = -1 \):**

   \[
   -2 \times (-1) = 2
   \]

### Final Answer:

\[
\lim_{{y \to \pi}} \frac{{\sin^2 y}}{{1 + \cos y}} = 2
\]

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Would you like more details or further clarification?

Here are 5 related questions for practice:

1. What is the limit \( \lim_{{x \to 0}} \frac{{\sin x}}{{x}} \)?
2. How do you apply L'Hôpital's Rule to \( \lim_{{x \to 0}} \frac{{\tan x}}{{x}} \)?
3. What happens to \( \lim_{{x \to 0}} \frac{{x^2}}{{1 - \cos x}} \)?
4. Evaluate \( \lim_{{x \to 0}} \frac{{1 - \cos x}}{{x^2}} \).
5. How does the squeeze theorem help solve limits involving trigonometric functions?

**Tip:** When you encounter an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), try L'Hôpital's Rule by differentiating the numerator and denominator separately.

Limit as y approaches pi of (sin^2(y)) / (1 + cos(y))

Find the limit of (sin^2(y)) / (1 + cos(y)) as y approaches pi using L'Hopital's Rule. This solution involves applying derivatives to resolve the indeterminate form 0/0 and requires knowledge of trigonometric identities.

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