https://file.aiquickdraw.com/mathbot/file/17264831750952b1mvl43.jpg

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.

Find lim (t → 13) (t² + 182) / (t² - 169)

This problem evaluates the limit of a rational function as t approaches 13. Through direct substitution, the denominator tends to zero, causing the function to approach infinity. Learn how to identify limits where the result does not exist. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation