If `(cos x)/(sin ax)` is periodic function, then
` lim_(mrarroo)(1+cos^(2m)n!pia)` is equal to

Options

A. 0
B. 1
C. 2
D. `-1`

Explanation

To solve the problem, we need to analyze the periodicity of the function \(\frac{\cos x}{\sin ax}\) and then evaluate the limit given in the question.

### Step-by-Step Solution:

1. **Understanding Periodicity**:
A function is periodic if it repeats its values at regular intervals. For the function \(\frac{\cos x}{\sin ax}\) to be periodic, \(a\) must be a rational number. This is because the sine function has a period of \(2\pi\), and for the quotient to also be periodic, the argument of the sine function must yield rational multiples of \(\pi\).

2. **Setting Up the Limit**:
We need to evaluate the limit:
\[ \lim_{m \to \infty} \left(1 + \cos^{2m} (n \pi a)\right) \]

3. **Analyzing \(\cos^{2m}(n \pi a)\)**:
The value of \(\cos(n \pi a)\) will depend on whether \(n \pi a\) is an integer multiple of \(\pi\) or not. If \(a\) is rational, say \(a = \frac{p}{q}\) where \(p\) and \(q\) are integers, then \(n \pi a = n \frac{p}{q} \pi\) will take values that are multiples of \(\pi\) for integer \(n\).

4. **Finding the Behavior of \(\cos^{2m}(n \pi a)\)**:
- If \(n \pi a\) is an integer multiple of \(\pi\), then \(\cos(n \pi a) = \cos(k \pi) = (-1)^k\) for some integer \(k\). Thus, \(\cos^{2m}(n \pi a) = 1\).
- If \(n \pi a\) is not an integer multiple of \(\pi\), then \(|\cos(n \pi a)| < 1\) and as \(m\) approaches infinity, \(\cos^{2m}(n \pi a)\) approaches \(0\).

5. **Evaluating the Limit**:
- If \(\cos(n \pi a) = 1\), then:
\[ \lim_{m \to \infty} \left(1 + \cos^{2m}(n \pi a)\right) = 1 + 1 = 2 \]
- If \(|\cos(n \pi a)| < 1\), then:
\[ \lim_{m \to \infty} \left(1 + \cos^{2m}(n \pi a)\right) = 1 + 0 = 1 \]

6. **Conclusion**:
Since \(a\) is rational, the limit will depend on the specific values of \(n\) and \(a\). However, since we are asked for the limit as \(m\) approaches infinity and considering the periodic nature of the function, the most general case leads us to conclude that the limit evaluates to \(2\) if \(n \pi a\) is an integer multiple of \(\pi\).

Thus, the final answer is:
\[ \boxed{2} \]