Classify the statement as either true or false.

If

ModifyingBelow lim With x right arrow 9

​F(x) 

​exists, then F

could be continuous
 
at

xequals

9

.

The statement is **true**.

Here's why:

If the limit of \( F(x) \) as \( x \) approaches 9 exists, then \( F(x) \) could potentially be continuous at \( x = 9 \). However, for \( F(x) \) to be continuous at \( x = 9 \), three conditions must be met:

1. \( F(9) \) must be defined.
2. \( \lim_{{x \to 9}} F(x) \) must exist.
3. \( \lim_{{x \to 9}} F(x) = F(9) \).

So, the existence of the limit \( \lim_{{x \to 9}} F(x) \) means that continuity is possible, but other conditions (like \( F(9) \) being defined and equal to the limit) must also be satisfied for \( F(x) \) to be continuous at \( x = 9 \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What are the formal conditions for a function to be continuous at a point?
2. Can a function be continuous if its limit exists but is not equal to its function value at the point?
3. How can we determine the existence of a limit at a point?
4. What is the difference between one-sided and two-sided limits?
5. Can a function have a limit at a point where it is not defined?

**Tip:** Continuity at a point requires the function's limit at that point to match the function's actual value at that point.

Classify the statement as either true or false. If \(\lim_{{x \to 9}} F(x)\) exists, then \(F\) could be continuous at \(x = 9\).

Explore the conditions under which a function can be considered continuous at a point. This article discusses the role of limits and the three key criteria for continuity, suitable for students in Grades 11-12.

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