(a) If f(2)=4, can you conclude anything about the limit of...

(a) If $f(2)=4,$ can you conclude anything about the limit of $f(x)$ as $x$ approaches 2$?$ Explain your reasoning.
(b) If the limit of $f(x)$ as $x$ approaches 2 is $4,$ can you conclude anything about $f(2) ?$ Explain your reasoning.

Key Concepts

Continuity and the Relationship Between Function Value and Limit

A function is continuous at a point if the limit as x approaches that point exists and is equal to the function's actual value at that point. Thus, if you know that the limit of f(x) as x approaches 2 is 4, it does not automatically mean that f(2) is defined or equals 4 unless the function is known to be continuous at 2. In the absence of continuity, a function can have a limit of 4 as x approaches 2 while f(2) can be different or even undefined.

Definition of a Limit

The limit of a function as x approaches a particular value is the value that the function approaches as x gets arbitrarily close to that point. This concept considers the behavior of the function near the point, not necessarily the value of the function at the point itself. Therefore, knowing f(2) = 4 only tells you the value of the function at that point and does not inherently provide information about the behavior of the function near 2 or the existence of a limit at 2.

Transcript

00:01 Okay, so here we have the following statement.

00:06 So if we have that some function evaluated at 2 is equals to 4, this implies that the limit of this function is equals to 4.

00:21 So that's the question here.

00:23 Is this true or not? so how do you should...

00:28 So we can either prove this or disprove by using some contours.

00:35 So let's use that.

00:36 So this is not true.

00:39 And i am going to show that.

00:41 So let's suppose that we have some function and let's define piecewise.

00:48 A piecewise function.

00:51 So it's going to be, for example, just for simplicity, it's going to be one.

01:00 If x is less than two.

01:03 Here we are going to say that it's two.

01:06 Sorry, here is four.

01:09 If x is equals to 2 and finally let's say that well is 5 when x is greater than 2 okay so we have this function and what happened well just by seeing here we can also plot this which make this the things easier so here we are going to put let me put a little to the left so here we have the plane and this function is going to be um let's say that here in some point here is two so here we have two okay so here we have one okay and to over this one here this is a black here there is a whole and for values great later than two, you have the following behavior.

02:33 We start that 5.

02:35 Let's say that is around here.

02:40 Let's say that 5 is here.

02:45 So here is 5.

02:51 And the function is like this.

02:54 And in some point on the middle, there is here, this point that is 4.

03:11 So here, we have a 4 point.

03:17 Okay, so this function, where at 2, this function is equal to 4, just as the statement say...