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The problem in your image asks to rigorously assert when the limit \(\lim_{x \to c} f(x) \neq L\) and to sequence the given phrases correctly.

To construct the rigorous definition for \(\lim_{x \to c} f(x) \neq L\), you need to invert the definition of a limit existing and equaling \(L\). The typical limit definition is:

\[
\lim_{x \to c} f(x) = L \iff \forall \epsilon > 0, \exists \delta > 0 \text{ such that } 0 < |x - c| < \delta \implies |f(x) - L| < \epsilon.
\]

To assert that the limit does **not** equal \(L\), you negate this. The negation would be:

\[
\exists \epsilon > 0, \forall \delta > 0, \exists x \text{ such that } 0 < |x - c| < \delta \text{ and } |f(x) - L| \geq \epsilon.
\]

This translates into ordering the phrases like this:

1. \(\exists \epsilon > 0\)  (Phrase 1)
2. \(\forall \delta > 0\)  (Phrase 2)
3. \(\exists x \text{ such that } 0 < |x - c| < \delta\)  (Phrase 8)
4. \(|f(x) - L| \geq \epsilon\)  (Phrase 4)
5. "but" indicating the negation context (Phrase 5)

So, the correct order for the rigorous assertion that \(\lim_{x \to c} f(x) \neq L\) is:

- 1
- 7
- 6
- 2
- 3
- 4
- 5
- 8

Let me know if you'd like to go deeper into this or have more questions!

Here are five related questions:
1. What does it mean for a limit to exist at a point?
2. How can we visualize \(\epsilon-\delta\) definitions graphically?
3. How do you negate a mathematical statement involving quantifiers?
4. How does this \(\epsilon-\delta\) definition apply to discontinuous functions?
5. How would you assert that a function diverges to infinity at a point using a similar framework?

**Tip:** The \(\epsilon-\delta\) definition is the foundation of calculus limits, and mastering it will help you understand the rigorous underpinnings of continuity and differentiability!

Recall that lim(x -> c) f(x) = L means: For all ε > 0 there is a δ > 0 such that for all x satisfying 0 < |x - c| < δ, we have that |f(x) - L| < ε. What if the limit does not equal L?

Explore how to rigorously assert when lim(x -> c) f(x) ≠ L by using the epsilon-delta definition. Learn how to negate the classical limit definition and understand the detailed steps to form this assertion, suitable for Calculus 1 students.

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