The value of `lim_(xrarroo) x^(2)(1-cos.(1)/(x))` is

Options

A. 0
B. `1//4`
C. `1//2`
D. 1

Explanation

To solve the limit \( \lim_{x \to \infty} x^2 (1 - \cos(1/x)) \), we can follow these steps:

### Step 1: Rewrite the expression using the half-angle identity
We know that \( 1 - \cos \theta = 2 \sin^2(\theta/2) \). Therefore, we can rewrite \( 1 - \cos(1/x) \) as:
\[ 1 - \cos(1/x) = 2 \sin^2(1/(2x)) \]

### Step 2: Substitute this into the limit
Now we substitute this back into our limit:
\[ \lim_{x \to \infty} x^2 (1 - \cos(1/x)) = \lim_{x \to \infty} x^2 \cdot 2 \sin^2(1/(2x)) \]
This simplifies to:
\[ = 2 \lim_{x \to \infty} x^2 \sin^2(1/(2x)) \]

### Step 3: Use the small angle approximation for sine
As \( x \to \infty \), \( \frac{1}{2x} \to 0 \). We can use the approximation \( \sin(y) \approx y \) when \( y \) is small. Thus:
\[ \sin(1/(2x)) \approx \frac{1}{2x} \]
So,
\[ \sin^2(1/(2x)) \approx \left(\frac{1}{2x}\right)^2 = \frac{1}{4x^2} \]

### Step 4: Substitute the approximation back into the limit
Substituting this approximation into our limit gives:
\[ 2 \lim_{x \to \infty} x^2 \cdot \frac{1}{4x^2} = 2 \cdot \frac{1}{4} = \frac{1}{2} \]

### Step 5: Conclusion
Thus, the value of the limit is:
\[ \lim_{x \to \infty} x^2 (1 - \cos(1/x)) = \frac{1}{2} \]

### Final Answer
The value of \( \lim_{x \to \infty} x^2 (1 - \cos(1/x)) \) is \( \frac{1}{2} \).

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