Use the given graph of f to state the value of each...

Use the given graph of $f$ to state the value of each quantity, if it exists. If it does not exist, explain why. $$ \begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 2^{-}} f(x)} & {\text { (b) } \lim _{x \rightarrow 2^{+}} f(x)} & {\text { (c) } \lim _{x \rightarrow 2} f(x)} \\ {\text { (d) } f(2)} & {\text { (e) } \lim _{x \rightarrow 4} f(x)} & {\text { (f) } f(4)}\end{array} $$

Use the given graph of $f$ to state the value of each quantity, if it exists. If it does not exist, explain why.
$$ \begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 2^{-}} f(x)} & {\text { (b) } \lim _{x \rightarrow 2^{+}} f(x)} & {\text { (c) } \lim _{x \rightarrow 2} f(x)} \\ {\text { (d) } f(2)} & {\text { (e) } \lim _{x \rightarrow 4} f(x)} & {\text { (f) } f(4)}\end{array} $$

Key Concepts

Graphical Analysis

Graphical analysis involves interpreting the behavior of a function by studying its graph. This method is used to visually assess limits and continuity, such as identifying open and closed circles at points, which indicate whether a function approaches a certain value or is defined at that point. It is especially useful in determining the existence of one-sided limits, the overall limit, and the actual function values.

One-Sided Limits

One-sided limits describe the behavior of a function as the input approaches a particular value from one side only—either from the left (values less than the point) or from the right (values greater than the point). These limits are crucial in analyzing functions graphically, especially when different behaviors are observed on either side of a point, which can indicate a jump or other types of discontinuities.

Limit of a Function

The limit of a function at a point captures the value that the function approaches as the input gets arbitrarily close to that point, regardless of the function's actual value at that point. For a two-sided limit to exist, both the left-hand and right-hand limits must exist and be equal.

Function Evaluation

Function evaluation refers to determining the value of a function at a specific input. This is distinct from finding a limit. A function's actual value at a point can differ from the limit as the input approaches that point, which is a key consideration when examining continuity and identifying removable discontinuities.

Continuity and Discontinuity

A function is continuous at a point if the limit as the input approaches the point equals the function value at that point. Discontinuities occur when this condition is not met; for example, if the left-hand and right-hand limits do not match (indicating a jump discontinuity) or if the limit exists but does not equal the function’s value (indicating a removable discontinuity).

Transcript

00:03 We are given the graph of a function, and we are given quantities, and we're asked to use the graph of this function to state the value of this quantity if it exists.

00:14 And if it does not exist, we're asked to explain why.

00:19 So the graph of the function, f, is featured in exercise 4 of this section.

00:29 And in part a, we're asked to find the quantity, the limit as x approaches 2 from the left, of f of x.

00:46 To find this limit, we'll look at the graph and we'll look to see how the graph behaves as we move towards x equals 2 from the left.

01:06 Well, we see that if we trace the graph with our finger or pencil, whatever we have, until we reach from left to right until we reach x equals 2, at this point, we see that f of the value of the function near this point is about 3.

01:36 Likewise, in part b, we're asked to find the limit as x approaches 2 from the right of f of x.

01:46 Similarly, we'll start to the right of x equals 2 on the graph, and then we'll trace with our finger or a pencil from right to left until i reach x equals 2.

02:01 If you do this, if you follow the graph until you reach x equals 2 from the right, you'll see that you end up at about the line y equals 1.

02:12 And so the value of the limit is 1.

02:15 Notice that the limit from part a and limit from part b have different values, even though we're approaching the same number on the x -axis, just from a different direction.

02:28 In part c, we're asked to find the limit as x approaches 2 of f of x if it exists.

02:36 Well, in fact, we know that this limit exists if and only if, the left -hand and right -hand limits exist.

02:55 Not only exist, but they also have to be equal.

03:01 However, we saw that in parts a and b, the limit as x approaches 2 from the left of f of x is not equal to the limit as x approaches 2 from the right of f of x...

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