The given limit is:

\[
\lim_{t \to 7} \frac{\sqrt{(t-7)^3}}{t-7}
\]

### Step-by-step solution:

1. **Rewrite the square root:**

   The expression \(\sqrt{(t-7)^3}\) can be rewritten as:

   \[
   \sqrt{(t-7)^3} = (t-7)^{3/2}
   \]

   So the limit becomes:

   \[
   \lim_{t \to 7} \frac{(t-7)^{3/2}}{t-7}
   \]

2. **Simplify the expression:**

   Now, simplify the fraction by subtracting exponents (since \((t-7)^{3/2} / (t-7) = (t-7)^{1/2}\)):

   \[
   \lim_{t \to 7} (t-7)^{1/2}
   \]

3. **Substitute the limit:**

   As \(t \to 7\), the term \((t - 7) \to 0\). So we are left with:

   \[
   \lim_{t \to 7} (t-7)^{1/2} = \sqrt{0} = 0
   \]

### Final Answer:

\[
\lim_{t \to 7} \frac{\sqrt{(t-7)^3}}{t-7} = 0
\]

Would you like further details on any of the steps? 

Here are 5 questions for more exploration:
1. What are common techniques for evaluating limits with radicals?
2. How do exponent rules apply when simplifying expressions involving limits?
3. How would this limit change if the exponent inside the square root were different?
4. What would happen if we approached a different value instead of 7?
5. How do limits behave when there's a higher degree of polynomial in the numerator?

**Tip:** Always simplify expressions involving radicals before substituting the limit value.

Discover how to evaluate the limit of the radical expression √(t-7)³ / (t-7) as t approaches 7. This problem explores the properties of limits and radical simplifications, suitable for students in 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