If `(5x-2)/x lt f(x) lt (5x^2-4x)/x^2` and `(sinx^2)/x lt g (x) lt (log (1+x^2))/x` then `|lim_( xtooo)f(x) -lim_( x to oo) g(x)|` is equal to

Answer

To solve the given problem, we need to evaluate the limits of the functions \( f(x) \) and \( g(x) \) as \( x \) approaches infinity, and then find the absolute difference between these two limits.

### Step 1: Evaluate \( \lim_{x \to \infty} f(x) \)

We are given:
\[ \frac{5x - 2}{x} < f(x) < \frac{5x^2 - 4x}{x^2} \]

First, simplify the bounds:

1. **Lower Bound**:
\[ \frac{5x - 2}{x} = 5 - \frac{2}{x} \]

2. **Upper Bound**:
\[ \frac{5x^2 - 4x}{x^2} = 5 - \frac{4}{x} \]

Now we have:
\[ 5 - \frac{2}{x} < f(x) < 5 - \frac{4}{x} \]

### Step 2: Take the limit as \( x \to \infty \)

Now, we evaluate the limits of the lower and upper bounds:
\[ \lim_{x \to \infty} \left(5 - \frac{2}{x}\right) = 5 - 0 = 5 \]
\[ \lim_{x \to \infty} \left(5 - \frac{4}{x}\right) = 5 - 0 = 5 \]

By the Squeeze Theorem (or Sandwich Theorem), since both bounds converge to 5, we conclude:
\[ \lim_{x \to \infty} f(x) = 5 \]

### Step 3: Evaluate \( \lim_{x \to \infty} g(x) \)

We are given:
\[ \frac{\sin(x^2)}{x} < g(x) < \frac{\log(1 + x^2)}{x} \]

1. **Lower Bound**:
\[ \frac{\sin(x^2)}{x} \]
As \( x \to \infty \), \( \sin(x^2) \) oscillates between -1 and 1, thus:
\[ -\frac{1}{x} < \frac{\sin(x^2)}{x} < \frac{1}{x} \]
Therefore, taking the limit:
\[ \lim_{x \to \infty} \frac{\sin(x^2)}{x} = 0 \]

2. **Upper Bound**:
\[ \frac{\log(1 + x^2)}{x} \]
As \( x \to \infty \), we can use the fact that \( \log(1 + x^2) \sim \log(x^2) = 2\log(x) \):
\[ \lim_{x \to \infty} \frac{\log(1 + x^2)}{x} = \lim_{x \to \infty} \frac{2\log(x)}{x} = 0 \]

By the Squeeze Theorem again:
\[ \lim_{x \to \infty} g(x) = 0 \]

### Step 4: Calculate \( | \lim_{x \to \infty} f(x) - \lim_{x \to \infty} g(x) | \)

Now we have:
\[ \lim_{x \to \infty} f(x) = 5 \quad \text{and} \quad \lim_{x \to \infty} g(x) = 0 \]
Thus:
\[ | \lim_{x \to \infty} f(x) - \lim_{x \to \infty} g(x) | = |5 - 0| = 5 \]

### Final Answer:
\[ \boxed{5} \]