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To find the limits of \( a = \sin^5 x + \cos^5 x \) for all values of \( x \), we need to analyze the range of this expression.

### Step-by-step Solution:

1. **Understand the Range of Sine and Cosine Functions:**
   - The sine and cosine functions, \( \sin(x) \) and \( \cos(x) \), have values ranging from -1 to 1.
   - Thus, \( \sin^5(x) \) and \( \cos^5(x) \) also range between -1 and 1 since raising to an odd power preserves the sign.

2. **Rewrite the Expression:**
   - Consider the expression:
     \[
     a = \sin^5(x) + \cos^5(x)
     \]
   - Notice that this expression can be factored using the identity:
     \[
     \sin^5(x) + \cos^5(x) = (\sin(x) + \cos(x))(\sin^4(x) - \sin^3(x) \cos(x) + \sin^2(x) \cos^2(x) - \sin(x) \cos^3(x) + \cos^4(x))
     \]

3. **Range of \( \sin(x) + \cos(x) \):**
   - Using the trigonometric identity, \( \sin(x) + \cos(x) = \sqrt{2} \sin(x + \pi/4) \), we know that \( \sin(x) + \cos(x) \) ranges between \( -\sqrt{2} \) and \( \sqrt{2} \).

4. **Finding the Minimum and Maximum Values:**
   - Since both \( \sin^5(x) \) and \( \cos^5(x) \) vary between -1 and 1, and knowing that \( \sin^5(x) + \cos^5(x) \) has to fall within that same interval, we conclude:
   - The **minimum value** occurs when one of the functions is -1 and the other is 0, which gives \( -1 \).
   - The **maximum value** occurs when one of the functions is 1 and the other is 0, which gives \( 1 \).

### Conclusion:

The limits of \( a = \sin^5(x) + \cos^5(x) \) for all values of \( x \) are:

\[
\boxed{[-1, 1]}
\]

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

### Related Questions:
1. What is the range of \( \sin^n(x) + \cos^n(x) \) for any positive integer \( n \)?
2. How would the range change if we consider \( \sin^4(x) + \cos^4(x) \)?
3. Can you derive the general expression for the range of \( a = \sin^n(x) + \cos^n(x) \) for even and odd \( n \)?
4. How does the identity \( \sin(x) + \cos(x) = \sqrt{2} \sin(x + \pi/4) \) help in analyzing trigonometric expressions?
5. What happens if we consider the expression \( \sin^6(x) + \cos^6(x) \)?

### Tip:
When dealing with trigonometric functions raised to a power, consider the fundamental trigonometric identities and their ranges to analyze the given expression efficiently.

If a = sin^5 x + cos^5 x, find the limits of a for all values of x.

Find the limits of the function a = sin^5 x + cos^5 x, using trigonometric principles. This problem explores the range of trigonometric functions raised to an odd power, showing that the limits are [-1, 1]. Suitable for Grades 9-12.

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