To solve the limit

$\lim_{t \to \infty} \frac{\sqrt{t + t^2}}{3t - t^2},$

let's simplify the expression step-by-step:

Step 1: Simplify the numerator

The numerator is $\sqrt{t + t^2}$. We can factor out $t^2$ inside the square root:

$\sqrt{t + t^2} = \sqrt{t^2(1 + \frac{t}{t^2})} = \sqrt{t^2(1 + \frac{1}{t})} = t\sqrt{1 + \frac{1}{t}}.$

As $t \to \infty$, $\frac{1}{t} \to 0$, so $\sqrt{1 + \frac{1}{t}} \to 1$. Thus, the numerator simplifies to approximately $t$.

Step 2: Simplify the denominator

The denominator is $3t - t^2$. Factoring out $t^2$ from the terms inside the expression:

$3t - t^2 = -t^2(1 - \frac{3}{t}).$

As $t \to \infty$, $\frac{3}{t} \to 0$, so the denominator simplifies to approximately $-t^2$.

Step 3: Combine the simplified expressions

Now, combine the simplified numerator and denominator:

$\frac{t\sqrt{1 + \frac{1}{t}}}{-t^2(1 - \frac{3}{t})} \approx \frac{t}{-t^2} = -\frac{1}{t}.$

Step 4: Take the limit

As $t \to \infty$, $-\frac{1}{t} \to 0$.

Therefore, the limit is:

$\boxed{0}.$

Would you like a detailed explanation of any part of this process, or do you have any further questions?

Here are some related questions that might interest you:

1. How do we handle limits that involve infinity and indeterminate forms?
2. What are some common techniques for simplifying expressions under square roots?
3. How can we use factoring to simplify complex fractions?
4. How do we find limits involving rational functions at infinity?
5. What are other techniques for evaluating limits involving square roots?

Tip: Always look for the highest power of the variable in the numerator and denominator when simplifying limits that approach infinity.