From the uploaded image, we are tasked to evaluate the limit of $f(x)$ as $x \to 0$ using the graph.

Step-by-step solution:

1. Observing the graph around $x = 0$:

   • As $x$ approaches $0$ from the left ($x \to 0^-$), $f(x)$ appears to approach $y = 1$.
   • Similarly, as $x$ approaches $0$ from the right ($x \to 0^+$), $f(x)$ also approaches $y = 1$.

2. Checking for continuity at $x = 0$:

   • Although there is a hole at $x = 0$ (indicating $f(0)$ is undefined), the limit depends only on the values of $f(x)$ as $x$ approaches $0$, not the actual value of $f(0)$.

3. Conclusion:

   • Since $f(x) \to 1$ from both sides of $x = 0$, we conclude that:

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

Would you like a deeper explanation or have additional questions?

Five Related Questions:

1. What is the formal definition of a limit, and how does it apply here?
2. How do left-hand and right-hand limits help determine overall limits?
3. Why doesn't the hole at $x = 0$ affect the limit?
4. Can a function have a limit at a point where it is not defined?
5. What is the difference between continuity and having a limit?

Tip:

To analyze limits graphically, always look for the behavior of the function on both sides of the target $x$-value, and ensure left-hand and right-hand limits match!