To solve these limit problems using the provided graph, let’s analyze the limits as $x$ approaches 0 from both sides for the function $f(x)$.

Observations from the Graph:

1. The graph of $y = f(x)$ shows that as $x \to 0^-$ (approaching from the left), the function $f(x)$ seems to approach a $y$-value of 2.
2. However, as $x \to 0^+$ (approaching from the right), the function $f(x)$ seems to decrease towards a different value below 1 (close to zero).

Limits:

• Left-Hand Limit $\lim_{x \to 0^-} f(x) = 2$
• Right-Hand Limit $\lim_{x \to 0^+} f(x) \approx 0$

Since the left-hand limit and right-hand limit are not equal, the two-sided limit does not exist at $x = 0$.

Solutions:

1. $\lim_{x \to 0^-} f(x) = 2$
2. $\lim_{x \to 0^+} f(x) = 0$
3. $\lim_{x \to 0} f(x)$ does not exist (DNE) because the left-hand and right-hand limits are not equal.

Would you like further details on the process?

Additional Questions:

1. How would the existence of the limit change if the left-hand and right-hand limits were equal?
2. What is the significance of the two-sided limit in terms of continuity at a point?
3. How can we determine limits graphically for more complex functions?
4. How might we estimate limits numerically if a graph is unavailable?
5. What are the conditions required for a function to have a limit at a point?

Tip:

Always check both the left-hand and right-hand limits separately when analyzing limits graphically to ensure the two-sided limit’s existence.