Let's examine the graph and determine if the limit $\lim_{x \to 0} f(x)$ exists, and if so, find its value.

Observations from the Graph:

1. As $x$ approaches 0 from the left ($x \to 0^-$):

   • The function $f(x)$ appears to approach the value 1.

2. As $x$ approaches 0 from the right ($x \to 0^+$):

   • The function $f(x)$ also appears to approach the value 1.

Conclusion:

Since both the left-hand limit and right-hand limit as $x$ approaches 0 are equal and approach the same value (1), the two-sided limit exists and is equal to 1.

Thus, $\lim_{x \to 0} f(x) = 1.$

Let me know if you'd like more details, or have any other questions!

Here are 5 related questions for further exploration:

1. What does it mean for a limit to not exist?
2. How do you evaluate limits algebraically versus graphically?
3. What is the formal definition of a limit (using epsilon-delta)?
4. How do limits relate to continuity at a point?
5. What happens when the left-hand and right-hand limits are not equal?

Tip: When both the left-hand and right-hand limits match at a point, the two-sided limit exists at that point.