Use the given graph of to state the value of each quantity,...

Use the given graph of to state the value of each quantity, if it exists. If it does not exist, explain why. $${ (a) }\lim _{x \rightarrow 1^{-}} f(x) \quad \text { (b) } \lim _{x \rightarrow 1^{+}} f(x) \quad \text { (c) } \lim _{x \rightarrow 1} f(x)$$ $${ (d) }\lim _{x \rightarrow 5} f(x) \quad \text { (e) } f(5)$$

Use the given graph of to state the value of each quantity, if it exists. If it does not exist, explain why.
$${ (a) }\lim _{x \rightarrow 1^{-}} f(x) \quad \text { (b) } \lim _{x \rightarrow 1^{+}} f(x) \quad \text { (c) } \lim _{x \rightarrow 1} f(x)$$
$${ (d) }\lim _{x \rightarrow 5} f(x) \quad \text { (e) } f(5)$$

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Key Concepts

Left-Hand Limit

The left-hand limit considers values of the function as the input approaches a specific value from values less than that point. It focuses exclusively on behavior coming from the left side of the point, which is crucial in understanding the function's behavior near points of potential discontinuity or jump.

Right-Hand Limit

The right-hand limit examines the behavior of the function as the input approaches a specific value from values greater than that point. This concept is key to determining if a function’s behavior is consistent from both sides of the point and to identifying potential discontinuities.

Two-Sided Limit

The two-sided limit exists when both the left-hand and right-hand limits exist and are equal as the input approaches a particular point. This concept is central in calculus for understanding overall continuity at a point, and it distinguishes between different types of discontinuities.

Relationship Between Limits and Function Values

This concept highlights that the limit of a function as the input approaches a point and the actual value of the function at that point can differ. Recognizing this distinction is important when analyzing graphs, particularly at points of removable discontinuities or jumps.

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