In Exercises 9 and 10, use the graph to find (i) limx →a^- f(x) (ii) limf(x),( iii) limf(x) , an

step 1 Okay, for this problem again, we want to find the limit at different points given the graph. The

In Exercises 9 and 10, use the graph to find (i) limx →a^-...

In Exercises 9 and $10,$ use the graph to find (i) $\lim _{x \rightarrow a^{-}} f(x)$
(ii) $\lim f(x),($ iii) $\lim f(x)$ , and $($ iv $) f(a)$ if it exists.
$${ (a) } a=1 \quad \text { (b) } a=2$$

Key Concepts

Function Value at a Point

The value of a function at a specific point is determined by its definition and indicates the actual output of the function at that input. It is important in contrast to the limit, as a function may have a limit at a point even if it is not defined there, leading to various types of discontinuities.

Continuity at a Point

A function is deemed continuous at a point if the overall limit as the input approaches that point exists, the function is defined at that point, and the limit equals the function's value at that point. This is a key concept in calculus that helps determine the smoothness and predictability of a function’s behavior.

Overall (Two-Sided) Limit

The overall or two-sided limit exists at a given point if and only if the left-hand and right-hand limits are equal. This concept underpins the fundamental definition of limits in calculus, providing insights into the behavior of a function as it approaches a particular input value from both sides.

Left-Hand Limit

The left-hand limit of a function at a point refers to the value that the function approaches as the input variable approaches that point from values less than the point. It is a crucial concept in understanding the behavior of functions near points of interest, especially when the function may behave differently on either side of that point.

Right-Hand Limit

The right-hand limit of a function at a point is the value the function approaches as the input variable approaches the point from values greater than the point. This concept is important for analyzing functions that could have different tendencies coming from the right side, and it is essential for verifying the overall limit at a point.

Transcript

Please give Ace some feedback