The image presents a question asking to explain the meaning of the limit:

$\lim_{{x \to a}} f(x) = L$

You are given three options:

• A: As $x$ approaches $a$ from either side, the value of $f(x)$ approaches $L$.
• B: As $x$ approaches $a$ from the right, the value of $f(x)$ approaches $L$.
• C: As $x$ approaches $a$ from the left, the value of $f(x)$ approaches $L$.

Explanation:

The expression $\lim_{{x \to a}} f(x) = L$ means that as $x$ gets arbitrarily close to $a$, the value of $f(x)$ approaches $L$. This limit definition does not specify whether $x$ approaches $a$ from the left or the right, meaning it considers both directions.

Therefore, the correct answer is A: As $x$ approaches $a$ from either side, the value of $f(x)$ approaches $L$.

Would you like more details or have any questions?

Related Questions:

1. What is the difference between one-sided and two-sided limits?
2. How do you evaluate limits when $f(x)$ approaches different values from the left and the right?
3. What does it mean if the two-sided limit does not exist?
4. How does the concept of continuity relate to limits?
5. What role do limits play in the definition of derivatives?

Tip: Always remember that a two-sided limit exists only if the function approaches the same value from both the left and right directions as $x$ approaches a particular point.