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.