`lim_(xto oo) (int_(0)^(x)(tan^(-1)t)^2dt)/(sqrt(x^(2)+1))` has the value
a)zero (b) `pi/4` (c) 1 (d) `(pi^2)/4`

Answer

To solve the limit

\[ L = \lim_{x \to \infty} \frac{\int_0^x (\tan^{-1} t)^2 dt}{\sqrt{x^2 + 1}}, \]

we will follow these steps:

### Step 1: Identify the form of the limit
As \( x \to \infty \), the integral in the numerator approaches infinity because the integrand \((\tan^{-1} t)^2\) is positive for \(t > 0\). The denominator \(\sqrt{x^2 + 1}\) also approaches infinity. Thus, we have an indeterminate form of \(\frac{\infty}{\infty}\).

**Hint:** Check the behavior of both the numerator and denominator as \(x\) approaches infinity to confirm the indeterminate form.

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that if \(\lim_{x \to c} \frac{f(x)}{g(x)}\) is of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then:

\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]

provided the limit on the right exists.

### Step 3: Differentiate the numerator and denominator
1. Differentiate the numerator:
\[ \frac{d}{dx} \left( \int_0^x (\tan^{-1} t)^2 dt \right) = (\tan^{-1} x)^2 \quad \text{(by the Fundamental Theorem of Calculus)} \]

2. Differentiate the denominator:
\[ \frac{d}{dx} \left( \sqrt{x^2 + 1} \right) = \frac{x}{\sqrt{x^2 + 1}}. \]

### Step 4: Rewrite the limit using derivatives
Now, we can rewrite the limit as:
\[ L = \lim_{x \to \infty} \frac{(\tan^{-1} x)^2}{\frac{x}{\sqrt{x^2 + 1}}} = \lim_{x \to \infty} \frac{(\tan^{-1} x)^2 \sqrt{x^2 + 1}}{x}. \]

### Step 5: Simplify the expression
As \(x\) approaches infinity:
- \(\tan^{-1} x\) approaches \(\frac{\pi}{2}\).
- \(\sqrt{x^2 + 1} \sim x\) (since for large \(x\), \(\sqrt{x^2 + 1} \approx x\)).

Substituting these into the limit gives:
\[ L = \lim_{x \to \infty} \frac{\left(\frac{\pi}{2}\right)^2 \cdot x}{x} = \left(\frac{\pi}{2}\right)^2 = \frac{\pi^2}{4}. \]

### Step 6: Final result
Thus, the value of the limit is:
\[ L = \frac{\pi^2}{4}. \]

### Conclusion
The answer is option (d) \(\frac{\pi^2}{4}\).

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