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Which of the following statements explains why the function is not continuous at x = 6?
f (6) is not defined
lim, →6- f(x) ≠ limx→6+ f(x)
lim f (x) ≠ f (6)
x→6

Looking at the provided graph of \( y = f(x) \), we need to assess the continuity at \( x = 6 \).

For a function to be continuous at a point \( x = c \), the following three conditions must be met:

1. **\( f(c) \) is defined**.
2. **\( \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) \)** (the left-hand limit equals the right-hand limit).
3. **The limit equals the function value**: \( \lim_{x \to c} f(x) = f(c) \).

### At \( x = 6 \):
- **\( f(6) \)** appears to be defined as a solid point, located at around \( y = 4 \).
- **The left-hand limit** \( \lim_{x \to 6^-} f(x) \) approaches \( y = 7 \) as \( x \) approaches 6 from the left.
- **The right-hand limit** \( \lim_{x \to 6^+} f(x) \) also approaches \( y = 7 \) as \( x \) approaches 6 from the right.

Thus, the limit exists and is equal to 7, but \( f(6) \neq 7 \). This breaks condition 3.

### Correct Explanation:
The correct statement is:

\[
\lim_{x \to 6} f(x) \neq f(6)
\]

Would you like further clarification on this? Here are 5 related questions to expand your understanding:

1. How can we formally define continuity for any function at a point?
2. What are other examples where a function is discontinuous because \( \lim f(x) \neq f(c) \)?
3. What happens to a function’s graph when the left-hand and right-hand limits don't match at a point?
4. What types of discontinuities exist, and how do they differ?
5. How does the behavior of limits help in identifying points of discontinuity?

**Tip**: Always remember, continuity issues at a point can arise due to limits or undefined values—it's essential to check both!

Which of the following statements explains why the function is not continuous at x = 6? f(6) is not defined; lim(x→6^-) f(x) ≠ lim(x→6^+) f(x); lim(x→6) f(x) ≠ f(6)

Learn how to determine the continuity of a function at x = 6 by analyzing limits and function values. This problem involves key concepts like left-hand and right-hand limits, as well as the definition of continuity. Suitable for students in Grades 10-12.

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