The problem shown is asking to compute the limit:

$\lim_{{t \to 13}} \frac{t^2 + 182}{t^2 - 169}.$

Let's work through this step by step:

Step 1: Substitute $t = 13$ directly

First, let's substitute $t = 13$ into the expression:

$t^2 = 13^2 = 169.$

Substitute into the numerator and denominator:

$\text{Numerator: } t^2 + 182 = 169 + 182 = 351,$ $\text{Denominator: } t^2 - 169 = 169 - 169 = 0.$

Step 2: Check the result

After substituting $t = 13$, we find that the denominator becomes 0, while the numerator becomes 351. This means the expression is approaching infinity (since you have a non-zero number divided by zero). Therefore, the limit does not exist because the function approaches infinity as $t$ approaches 13.

Conclusion:

The correct answer is Option B: The limit does not exist.

Would you like more details on this? Also, here are 5 related questions you could explore:

1. What happens when both the numerator and the denominator approach zero?
2. How do we handle limits when the expression tends towards infinity?
3. What is the concept of limits in calculus and why are they important?
4. Can we apply L'Hopital's Rule to limits like this? Why or why not?
5. What are vertical asymptotes and how do they relate to limits?

Tip: When evaluating limits, always check if direct substitution leads to indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, which may require further steps like factoring or using L'Hopital's Rule.