Evaluate the following limit. (Round your answer to three decimal places. If an answer does not exist, enter DNE.)

lim *h*→0 

6*h* − 1

*h*

= 

To evaluate the limit

\[
\lim_{h \to 0} \frac{6h - 1}{h},
\]

we can start by rewriting the expression:

\[
\lim_{h \to 0} \frac{6h - 1}{h} = \lim_{h \to 0} \left(\frac{6h}{h} - \frac{1}{h}\right).
\]

Simplifying this expression:

\[
\lim_{h \to 0} \left(6 - \frac{1}{h}\right).
\]

### Step 1: Analyze the limit components
- The term \( 6 \) is constant and does not depend on \( h \).
- The term \( \frac{1}{h} \) becomes undefined as \( h \) approaches 0. Specifically, as \( h \to 0^+ \) (from the right), \( \frac{1}{h} \to +\infty \). As \( h \to 0^- \) (from the left), \( \frac{1}{h} \to -\infty \).

### Step 2: Conclusion
Since \( \frac{1}{h} \) approaches \( +\infty \) and \( -\infty \) depending on the direction from which \( h \) approaches zero, the overall limit does not converge to a finite number. Instead, it becomes unbounded.

Therefore, the limit does not exist (DNE).

\[
\lim_{h \to 0} \frac{6h - 1}{h} = \text{DNE}.
\]

Would you like more details, or do you have any other questions? Here are some related questions you might find interesting:

1. What happens if the limit is evaluated as a one-sided limit from the left or right?
2. How do we handle similar limits with different forms, such as \( \lim_{h \to 0} \frac{h^2 - 1}{h} \)?
3. What does it mean for a limit to not exist? Are there different types of non-existence?
4. How does this relate to the concept of vertical asymptotes in graphing?
5. Can L'Hôpital's Rule be applied to limits like this? Why or why not?

**Tip:** When dealing with limits, always consider both one-sided limits to understand the behavior of the function near the point of interest.

Learn how to evaluate the limit lim_{h \to 0} (6h - 1) / h. Understand why the limit does not exist (DNE) due to the behavior of the expression as h approaches 0. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation