let [x] denote the greatest integer less than or equal to x.
Then `lim_(xto0) (tan(pisin^2x)+(abs.x-sin(x[x]))^2)/x^2`

Options

A. equals `pi`
B. equals 0
C. equals `pi+1`
D. does not exist

Answer

To solve the limit

\[ \lim_{x \to 0} \frac{\tan(\pi \sin^2 x) + (|x| - \sin(x[x]))^2}{x^2}, \]

we will analyze the expression step by step.

### Step 1: Analyze the term \(\tan(\pi \sin^2 x)\)

As \(x \to 0\), we know that \(\sin x \approx x\). Therefore,

\[ \sin^2 x \approx x^2. \]

This gives us:

\[ \tan(\pi \sin^2 x) \approx \tan(\pi x^2). \]

Using the small angle approximation \(\tan y \approx y\) when \(y\) is small, we have:

\[ \tan(\pi x^2) \approx \pi x^2. \]