The value of `lim_(mtooo) ("cos"(x)/(m))^(m)` is

Options

A. `underset(xtooo)lim(log_(e)x)/({x})=oo`
B. `underset(xto2^(+))lim(x)/(x^(2)-x-2)=oo`
C. `underset(xto-1^(-))lim(x)/(x^(2)-x-2)=oo`
D. `underset(xtooo)lim(log_(0.5)x)/({x})=oo`

Explanation

To solve the limit \( \lim_{m \to \infty} \left( \frac{\cos x}{m} \right)^m \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the limit:
\[ \lim_{m \to \infty} \left( \frac{\cos x}{m} \right)^m \]
As \( m \) approaches infinity, \( \frac{\cos x}{m} \) approaches \( 0 \) since \( m \) grows without bound. Therefore, we have a form that resembles \( 0^m \), which is indeterminate.

### Step 2: Transform the limit
We can rewrite the expression using the exponential function:
\[ \left( \frac{\cos x}{m} \right)^m = e^{m \ln\left(\frac{\cos x}{m}\right)} = e^{m (\ln(\cos x) - \ln(m))} \]
Thus, we need to evaluate:
\[ \lim_{m \to \infty} m (\ln(\cos x) - \ln(m)) \]

### Step 3: Analyze the limit
As \( m \to \infty \):
- \( \ln(m) \) approaches infinity.
- \( \ln(\cos x) \) is a constant (assuming \( \cos x \neq 0 \)).

Thus, we can simplify:
\[ m (\ln(\cos x) - \ln(m)) = m \ln(\cos x) - m \ln(m) \]
The term \( -m \ln(m) \) will dominate as \( m \to \infty \).

### Step 4: Evaluate the limit
We can express the limit as:
\[ \lim_{m \to \infty} (m \ln(\cos x) - m \ln(m)) = \lim_{m \to \infty} m \left(\ln(\cos x) - \ln(m)\right) \]
Since \( \ln(m) \) grows without bound, the entire limit approaches \( -\infty \).

### Step 5: Conclude the limit
Thus, we have:
\[ \lim_{m \to \infty} m (\ln(\cos x) - \ln(m)) = -\infty \]
This implies:
\[ e^{-\infty} = 0 \]

### Final Answer
Therefore, the value of the limit is:
\[ \lim_{m \to \infty} \left( \frac{\cos x}{m} \right)^m = 0 \]

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