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The given problem asks for the limit:

\[
\lim_{x \to -5} \cos(-5\pi x)
\]

### Step-by-step solution:

1. **Substitute \( x = -5 \) into the expression**:
   \[
   \cos(-5\pi (-5)) = \cos(25\pi)
   \]

2. **Simplify \( \cos(25\pi) \)**:
   The cosine function is periodic with a period of \( 2\pi \), so:
   \[
   \cos(25\pi) = \cos(25\pi \mod 2\pi) = \cos(\pi)
   \]

3. **Evaluate \( \cos(\pi) \)**:
   \[
   \cos(\pi) = -1
   \]

Thus, the limit is:

\[
\lim_{x \to -5} \cos(-5\pi x) = -1
\]

Would you like any more details or further clarifications?

### Related questions:
1. What is the general periodicity of the cosine function?
2. How would this problem change if we had a sine function instead of cosine?
3. Can you explain why \( \cos(25\pi) = \cos(\pi) \) due to periodicity?
4. How does the behavior of cosine function differ for even and odd multiples of \( \pi \)?
5. What would happen if the argument inside cosine was an irrational multiple of \( \pi \)?

### Tip:
When dealing with trigonometric limits, always try to simplify the argument using the periodic properties of sine and cosine functions to reduce it within the interval \( [0, 2\pi] \).

Explore the step-by-step solution to find the limit of cos(-5πx) as x approaches -5. This problem covers important concepts such as limits, the periodicity of trigonometric functions, and the evaluation of cosine function values. Ideal for students in Grades 11-12.

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