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To solve this problem, let’s analyze the given conditions and required expression.

1. We are given:
   \[
   \lim_{N \to \infty} \int_0^N e^{-t^2} \, dt = \frac{\sqrt{\pi}}{2}.
   \]

2. We need to evaluate:
   \[
   \lim_{N \to \infty} \int_0^N \frac{1}{t^2} \left( e^{-a t^2} - e^{-b t^2} \right) dt.
   \]

### Step-by-Step Solution Outline
This expression suggests an application of properties related to Gaussian integrals and possibly differentiation under the integral sign. By carefully handling each part, we aim to reduce the expression and match it to one of the options provided.

Would you like a detailed step-by-step solution for this problem? 

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### Follow-up Questions:
1. How does the Gaussian integral property apply to this problem?
2. What is the significance of \(a < b\) in the integral evaluation?
3. How does differentiating under the integral sign work for Gaussian integrals?
4. What role does the limit as \(N \to \infty\) play in these integrals?
5. Could there be alternative methods to solve integrals of this type?

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#### Tip:
For integrals involving exponential functions, consider using Gaussian integral results or series expansion techniques.

Let a, b be positive real numbers such that a < b. Given that \( \lim_{N \to \infty} \int_0^N e^{-t^2} dt = \frac{\sqrt{\pi}}{2} \), the value of \( \lim_{N \to \infty} \int_0^N \frac{1}{t^2} (e^{-a t^2} - e^{-b t^2}) dt \) is equal to?

This problem involves evaluating the limit of an integral with exponential terms and parameters, applying Gaussian integral results. It is suitable for students with an understanding of advanced calculus, limits, and integrals.

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