If `f(x)=lim_(nrarroo) (cos(x)/(sqrtn))^(n)`, then the value of `lim_(xrarr0) (f(x)-1)/(x)` is

Options

A. 0
B. 1
C. 2
D. `3//2`

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Understand the function \( f(x) \)
We are given:
\[ f(x) = \lim_{n \to \infty} \left( \frac{\cos(x)}{\sqrt{n}} \right)^n \]
As \( n \) approaches infinity, we need to analyze the expression inside the limit.

**Hint:** Identify the form of the limit as \( n \to \infty \).

### Step 2: Analyze the limit
As \( n \to \infty \), \( \sqrt{n} \) also approaches infinity. Thus, \( \frac{\cos(x)}{\sqrt{n}} \) approaches \( 0 \) for any \( x \neq 0 \). The expression \( \left( \frac{\cos(x)}{\sqrt{n}} \right)^n \) takes the form \( 0^n \), which approaches \( 0 \).

**Hint:** Consider the behavior of \( \cos(x) \) as \( n \) becomes very large.

### Step 3: Special case when \( x = 0 \)
When \( x = 0 \):
\[ f(0) = \lim_{n \to \infty} \left( \frac{\cos(0)}{\sqrt{n}} \right)^n = \lim_{n \to \infty} \left( \frac{1}{\sqrt{n}} \right)^n = \lim_{n \to \infty} \frac{1}{n^{n/2}} = 0 \]

**Hint:** Evaluate \( f(x) \) at \( x = 0 \) to see if it changes the limit.

### Step 4: General case for \( f(x) \)
For \( x \neq 0 \), we have:
\[ f(x) = 0 \]
Thus, we can conclude:
\[ f(x) = 0 \text{ for all } x \neq 0 \]
And \( f(0) = 0 \).

**Hint:** Confirm that \( f(x) \) is continuous at \( x = 0 \).

### Step 5: Evaluate the limit \( \lim_{x \to 0} \frac{f(x) - 1}{x} \)
Now we need to find:
\[ \lim_{x \to 0} \frac{f(x) - 1}{x} \]
Since \( f(x) = 0 \) for \( x \neq 0 \):
\[ \lim_{x \to 0} \frac{0 - 1}{x} = \lim_{x \to 0} \frac{-1}{x} \]
This limit does not exist as it approaches \( -\infty \) when \( x \) approaches \( 0 \) from the right and \( +\infty \) when approaching from the left.

**Hint:** Analyze the limit behavior as \( x \) approaches \( 0 \) from both sides.

### Final Answer
The limit \( \lim_{x \to 0} \frac{f(x) - 1}{x} \) does not exist (it diverges).