The value of: `lim_(n to infty) cos(x/2) cos(x/4) cos(x/8)….. Cos(x/2^(n))` is:

Options

A. 1
B. `(sin x)/x`
C. `x/(sin x)`
D. none of these

Answer

To solve the limit \( \lim_{n \to \infty} \cos\left(\frac{x}{2}\right) \cos\left(\frac{x}{4}\right) \cos\left(\frac{x}{8}\right) \cdots \cos\left(\frac{x}{2^n}\right) \), we can follow these steps:

### Step 1: Rewrite the Product
We can express the product of cosines in a more manageable form. The limit can be rewritten as:
\[ P_n = \prod_{k=1}^{n} \cos\left(\frac{x}{2^k}\right) \]

### Step 2: Use the Identity for Cosine
We use the identity \( \cos(a) = \frac{\sin(2a)}{2\sin(a)} \) to express each cosine term:
\[ \cos\left(\frac{x}{2^k}\right) = \frac{\sin\left(\frac{x}{2^{k-1}}\right)}{2\sin\left(\frac{x}{2^k}\right)} \]
Thus, we can express \( P_n \) as:
\[ P_n = \frac{\sin\left(\frac{x}{2^0}\right)}{2\sin\left(\frac{x}{2^1}\right)} \cdot \frac{\sin\left(\frac{x}{2^1}\right)}{2\sin\left(\frac{x}{2^2}\right)} \cdots \frac{\sin\left(\frac{x}{2^{n-1}}\right)}{2\sin\left(\frac{x}{2^n}\right)} \]

### Step 3: Simplify the Expression
Notice that in the product, all terms except the first sine and the last sine cancel out:
\[ P_n = \frac{\sin\left(x\right)}{2^n \sin\left(\frac{x}{2^n}\right)} \]

### Step 4: Take the Limit as \( n \to \infty \)
Now we need to evaluate:
\[ \lim_{n \to \infty} P_n = \lim_{n \to \infty} \frac{\sin(x)}{2^n \sin\left(\frac{x}{2^n}\right)} \]
As \( n \to \infty \), \( \frac{x}{2^n} \to 0 \), and we can use the limit property \( \lim_{y \to 0} \frac{\sin(y)}{y} = 1 \):
\[ \sin\left(\frac{x}{2^n}\right) \sim \frac{x}{2^n} \quad \text{as } n \to \infty \]
Thus,
\[ \lim_{n \to \infty} \sin\left(\frac{x}{2^n}\right) = \frac{x}{2^n} \]

### Step 5: Final Calculation
Substituting this back into our limit gives:
\[ \lim_{n \to \infty} P_n = \lim_{n \to \infty} \frac{\sin(x)}{2^n \cdot \frac{x}{2^n}} = \lim_{n \to \infty} \frac{\sin(x)}{x} = \frac{\sin(x)}{x} \]

### Conclusion
Therefore, the value of the limit is:
\[ \lim_{n \to \infty} \cos\left(\frac{x}{2}\right) \cos\left(\frac{x}{4}\right) \cos\left(\frac{x}{8}\right) \cdots = \frac{\sin(x)}{x} \]