find each limit for f(x)= square root x-6  lim f(x) x-> 6-

To find the left-hand limit of the function \( f(x) = \sqrt{x - 6} \) as \( x \to 6^- \), let's examine the behavior of the square root function near 6 from the left.

### Function and Limit Setup:
We are asked to find:

\[
\lim_{x \to 6^-} \sqrt{x - 6}
\]

### Behavior of \( f(x) \) as \( x \to 6^- \):
- The function \( f(x) = \sqrt{x - 6} \) is defined only for \( x \geq 6 \), since the square root of a negative number is not a real number.
- As \( x \to 6^- \) (from values less than 6), \( x - 6 \) becomes negative, which means \( \sqrt{x - 6} \) is not defined for real numbers.
- Therefore, the limit does not exist in the real number system.

### Conclusion:
\[
\lim_{x \to 6^-} \sqrt{x - 6} = \text{Does not exist (DNE)}
\]

Would you like further clarification or additional details? 

Here are 5 related questions to consider:
1. What is the right-hand limit of \( f(x) = \sqrt{x - 6} \) as \( x \to 6^+ \)?
2. How do we evaluate limits for functions involving square roots near undefined points?
3. What is the domain of the function \( f(x) = \sqrt{x - 6} \)?
4. Can we find a limit involving \( f(x) = \sqrt{x - 6} \) as \( x \to \infty \)?
5. How do we handle limits of piecewise functions near points of discontinuity?

**Tip**: Always check the domain of a function before calculating limits to ensure the function is defined near the point of interest.

Understand the behavior of the limit for f(x) = sqrt(x - 6) as x approaches 6 from the left. This problem focuses on limits, square root functions, and how limits behave near undefined points. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation