We are given a function $f$ that is continuous and strictly positive on the interval $[0, 1]$, and asked to compute the following limit:

$\lim_{x \to 0^+} x \int_{x}^{1} \frac{f(t)}{t^2} \, dt$

Step-by-Step Solution:

1. Rewrite the expression:
   We need to evaluate the limit of the product of $x$ and the integral:

   $x \int_{x}^{1} \frac{f(t)}{t^2} \, dt$

2. Look for the behavior as $x \to 0^+$:
   Since $f(t)$ is continuous and strictly positive on $[0, 1]$, it implies that as $x \to 0^+$, $f(t)$ will behave nicely and will not approach zero or infinity in an unbounded way. Let's break down the behavior of the integrand $\frac{f(t)}{t^2}$ near $t = 0$.

3. Estimate the integral:
   Since $f(t)$ is continuous on $[0, 1]$, $f(t)$ is bounded, meaning there exists a constant $M > 0$ such that for all $t \in [0, 1]$, $f(t) \leq M$. Therefore, we can estimate the integral:

   $\int_{x}^{1} \frac{f(t)}{t^2} \, dt \leq \int_{x}^{1} \frac{M}{t^2} \, dt$

4. Evaluate the integral:
   The integral $\int \frac{1}{t^2} \, dt = -\frac{1}{t}$, so:

   $\int_{x}^{1} \frac{M}{t^2} \, dt = M \left( -\frac{1}{t} \right) \Big|_x^1 = M \left( -\frac{1}{1} + \frac{1}{x} \right) = M \left( \frac{1}{x} - 1 \right)$

5. Combine with the factor $x$:
   Now, multiply this by $x$:

   $x \int_{x}^{1} \frac{f(t)}{t^2} \, dt \leq x M \left( \frac{1}{x} - 1 \right) = M \left( 1 - x \right)$

6. Take the limit as $x \to 0^+$:
   As $x \to 0^+$, $M(1 - x) \to M$. Therefore, the integral behaves like:

   $\lim_{x \to 0^+} x \int_{x}^{1} \frac{f(t)}{t^2} \, dt = M$

7. Conclusion:
   Since $f(t)$ is continuous and strictly positive, we conclude that the limit is $M$, the maximum value of $f(t)$ as $t$ approaches 0.

Thus, the final result is:

$\boxed{M}$

where $M$ is a constant such that $f(t)$ is bounded by $M$ over the interval $[0, 1]$.

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Do you need further details or clarifications on any steps?

Here are 5 related questions that can expand your understanding:

1. How can the continuity and positivity of $f(t)$ affect the behavior of the integral in different scenarios?
2. What is the significance of the bound $M$ in this problem?
3. Can you derive the asymptotic behavior of $x \int_{x}^{1} \frac{f(t)}{t^2} \, dt$ for different types of functions $f(t)$?
4. How does the choice of the limits in the integral influence the result when taking the limit as $x \to 0^+$?
5. What happens to the integral if $f(t)$ has singularities or is not bounded near 0?

Tip: When faced with limits involving integrals, consider bounding the integrand to estimate its behavior, especially if the function involved is continuous or has known properties over the interval.