For the function f whose graph is given, state the value of...

For the function $f$ whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. $$ \text { (a) }\lim _{x \rightarrow 1} f(x) \quad \text { (b) } \lim _{x \rightarrow 3^{-}} f(x) \quad \text { (c) } \lim _{x \rightarrow 3^{+}} f(x) $$ $$ \text { (d) } \lim _{x \rightarrow 3} f(x) \quad \text { (e) } f(3) $$

For the function $f$ whose graph is given, state the value of
each quantity, if it exists. If it does not exist, explain why.
$$
\text { (a) }\lim _{x \rightarrow 1} f(x) \quad \text { (b) } \lim _{x \rightarrow 3^{-}} f(x) \quad \text { (c) } \lim _{x \rightarrow 3^{+}} f(x)
$$
$$
\text { (d) } \lim _{x \rightarrow 3} f(x) \quad \text { (e) } f(3)
$$

Key Concepts

Function Value versus Limit

This concept highlights the distinction between the limit of a function as it approaches a point and the actual value of the function at that point. While in many cases (continuous functions) they are equal, there are notable situations, such as with removable or jump discontinuities, where the limit exists but does not equal the function’s value, or where the function’s value is undefined.

Existence of Two-Sided Limits

For a two-sided limit of a function at a point to exist, the one-sided limits (from the left and from the right) must both exist and be equal. If they differ, the overall limit does not exist at that point. This concept is crucial for understanding the overall behavior of functions and is also a key condition for discussing continuity.

One-Sided Limits

One-sided limits refer to the values that a function approaches as the variable approaches a particular point from only one side – either from the left (denoted as the left-hand limit) or the right (denoted as the right-hand limit). They are particularly important when a function has different behaviors on either side of a point, such as at points of discontinuity or at the boundaries of a function’s domain.

Limit of a Function

This concept involves determining the value a function approaches as the input gets arbitrarily close to a certain value. It is central to calculus because it forms the foundation for defining derivatives and integrals. The limit captures the behavior of the function near a specific point, even if the function is not actually defined at that point.

Transcript

00:01 So this problem has five parts.

00:03 We're obviously going to start with part a.

00:06 And what we can do is just look at the graph of something like this.

00:12 And we have this point in one comment two.

00:16 On the graph.

00:17 And to find the limit as x approaches 1 of f of x, all we have to do is look at this point.

00:23 And we see that there are no discontinuities, that is holes, step functions, asymptotes.

00:30 Nothing like that.

00:31 Here and we said that the limit as x approaches one from the left is equal to the limit as x approaches one from the right and therefore the limit as x approaches one of f of x is simply just so now for parts b through e we're going they all deal with the limit as x approaches three so we'll look at this graph looks like this where these points are three comma one three 3, 3, and 3 comma 4.

01:12 So we'll reference this for parts b through e.

01:16 Part b is simply asking for the limit as x approaches 3 from the left.

01:21 And if we track f of x graph, we can see that as f of x gets infinitely close to 3, it approaches is 0 .3, 1.

01:32 And so that's all the answer is going to be there.

01:34 The limit as x approaches 3 from the left of f of x is equal to 1.

01:43 Now for a part of c, it's going to deal similarly with the right side.

01:48 So we're looking for the limit as x approaches 3 from the right...

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