What is true if $$ f(x) $$ is continuous at $$ x=4 $$ and $$ f(4)=3 ? $$

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If $$ f(x) $$ is continuous at $$ x=4 $$ and $$ f(4)=3 $$, the following statements are true:

1. **Limit Condition**: The limit of $$ f(x) $$ as $$ x $$ approaches 4 must equal the value of the function at that point. Mathematically, this can be expressed as:
   $$
   \lim_{x \to 4} f(x) = f(4)
   $$
   Since $$ f(4) = 3 $$, we have:
   $$
   \lim_{x \to 4} f(x) = 3
   $$

2. **Existence of Limit**: The limit $$ \lim_{x \to 4} f(x) $$ must exist. This means that as $$ x $$ gets arbitrarily close to 4 from both the left and the right, the values of $$ f(x) $$ must approach the same number, which is 3 in this case.

3. **No Jump or Asymptote**: The function does not have any jumps, breaks, or asymptotes at $$ x=4 $$. This is a key characteristic of continuity.

In summary, the continuity of $$ f(x) $$ at $$ x=4 $$ implies that the function approaches the same value (3) from both sides as $$ x $$ approaches 4, and that there are no discontinuities at that point.
If $$ f(x) $$ is continuous at $$ x=4 $$ and $$ f(4)=3 $$, then: 1. The limit of $$ f(x) $$ as $$ x $$ approaches 4 is 3. 2. The function approaches 3 from both sides as $$ x $$ gets close to 4. 3. There are no jumps or breaks at $$ x=4 $$. In other words, $$ f(x) $$ smoothly reaches 3 when $$ x $$ is near 4.

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