The value of `lim_(nrarroo)(cos.(x)/(2)cos.(x)/(4)cos.(x)/(8)………cos.(x)/(2^(n+1)))` is equal to

Options

A. `(x)/(sinx)`
B. `(sinx)/(x)`
C. 0
D. None of these

Answer

To solve the limit

\[ \lim_{n \to \infty} \left( \cos\left(\frac{x}{2}\right) \cos\left(\frac{x}{4}\right) \cos\left(\frac{x}{8}\right) \cdots \cos\left(\frac{x}{2^{n+1}}\right) \right), \]

we can follow these steps:

### Step 1: Rewrite the Product
We can express the product in a more manageable form:

\[ \lim_{n \to \infty} \prod_{k=1}^{n+1} \cos\left(\frac{x}{2^k}\right). \]

### Step 2: Use the Identity for Sine
Recall the identity for sine:

\[ \sin(2x) = 2 \sin(x) \cos(x). \]

We can apply this identity iteratively. For example,

\[ \sin\left(\frac{x}{2}\right) = 2 \sin\left(\frac{x}{4}\right) \cos\left(\frac{x}{4}\right). \]

By applying this identity repeatedly, we can express the sine function in terms of cosines.

### Step 3: Express Sine in Terms of Cosine
Continuing from our previous identity, we can write:

\[ \sin\left(\frac{x}{2^k}\right) = 2 \sin\left(\frac{x}{2^{k+1}}\right) \cos\left(\frac{x}{2^{k+1}}\right). \]

If we apply this for all \( k \) from 1 to \( n \), we can express \( \sin(x) \) as:

\[ \sin(x) = 2^n \sin\left(\frac{x}{2^n}\right) \prod_{k=1}^{n} \cos\left(\frac{x}{2^k}\right). \]

### Step 4: Take the Limit
Now, taking the limit as \( n \to \infty \):

\[ \lim_{n \to \infty} \prod_{k=1}^{n} \cos\left(\frac{x}{2^k}\right) = \frac{\sin(x)}{2^n \sin\left(\frac{x}{2^n}\right)}. \]

As \( n \to \infty \), \( \frac{x}{2^n} \to 0 \), and we can use the limit property \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \):

\[ \lim_{n \to \infty} \frac{\sin\left(\frac{x}{2^n}\right)}{\frac{x}{2^n}} = 1. \]

Thus, we have:

\[ \lim_{n \to \infty} \prod_{k=1}^{n} \cos\left(\frac{x}{2^k}\right) = \frac{\sin(x)}{x}. \]

### Final Result
Therefore, the limit evaluates to:

\[ \lim_{n \to \infty} \left( \cos\left(\frac{x}{2}\right) \cos\left(\frac{x}{4}\right) \cos\left(\frac{x}{8}\right) \cdots \cos\left(\frac{x}{2^{n+1}}\right) \right) = \frac{\sin(x)}{x}. \]