Statement 1: If `f(x)=2/(pi) cot^(-1)((3x^(2)+1)/((x-1)(x-2)))`, then `lim_(xto1^(-))f(x)=0` and `lim_(xto2^(-))f(x)=2`
Statement 2: `lim_(xtooo)cot^(-1)x=0` and `lim_(xto -oo)cot^(-1)x=pi`

Options

A. Statement-1 is True, Statement -2 is True, Statement -2 is True and Statement -2 is a correct explanation for statement -1.
B. Statement -1 is True, Statement -2 is True and Statement -2 is Not is a correct explanation for Statement -1.
C. Statement -1 is True, Statement -2 is False.
D. Statement -1 is False, Statement -2 is True.

Answer

To solve the given problem, we will analyze both statements step by step.

### Step 1: Analyze Statement 1

We have the function:
\[ f(x) = \frac{2}{\pi} \cot^{-1}\left(\frac{3x^2 + 1}{(x-1)(x-2)}\right) \]

#### Step 1.1: Find \(\lim_{x \to 1^-} f(x)\)

As \(x\) approaches \(1\) from the left:
- The expression \((x-1)(x-2)\) approaches \(0\) (specifically, it approaches \(0\) from the negative side since \(x < 1\)).
- Therefore, \(\frac{3x^2 + 1}{(x-1)(x-2)}\) approaches \(-\infty\).

Now, we know:
\[ \cot^{-1}(-\infty) = \pi \]
Thus,
\[ \lim_{x \to 1^-} f(x) = \frac{2}{\pi} \cdot \pi = 2 \]

#### Step 1.2: Find \(\lim_{x \to 2^-} f(x)\)

As \(x\) approaches \(2\) from the left:
- The expression \((x-1)(x-2)\) again approaches \(0\) (specifically, it approaches \(0\) from the negative side).
- Therefore, \(\frac{3x^2 + 1}{(x-1)(x-2)}\) approaches \(-\infty\).

Thus,
\[ \lim_{x \to 2^-} f(x) = \frac{2}{\pi} \cdot \pi = 2 \]

### Conclusion for Statement 1:
- \(\lim_{x \to 1^-} f(x) = 2\) (not 0 as claimed)
- \(\lim_{x \to 2^-} f(x) = 2\) (correct)

Thus, Statement 1 is **false**.

### Step 2: Analyze Statement 2

#### Step 2.1: Find \(\lim_{x \to \infty} \cot^{-1}(x)\)

As \(x\) approaches \(+\infty\):
\[ \cot^{-1}(x) \to 0 \]

#### Step 2.2: Find \(\lim_{x \to -\infty} \cot^{-1}(x)\)

As \(x\) approaches \(-\infty\):
\[ \cot^{-1}(x) \to \pi \]

### Conclusion for Statement 2:
Both limits are correct:
- \(\lim_{x \to \infty} \cot^{-1}(x) = 0\)
- \(\lim_{x \to -\infty} \cot^{-1}(x) = \pi\)

Thus, Statement 2 is **true**.

### Final Conclusion:
- Statement 1 is false.
- Statement 2 is true.