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Problem 1 Easy Difficulty

Explain in your own words what is meant by the equation $ \displaystyle \lim_{x\to 2} f(x) = 5 $
Is it possible for this statement to be true and yet $ f(2) = 3 $?
Explain.

Answer

As $x$ approaches $2, f(x)$ approaches $5 .[\mathrm{Or}, \text { the values of } f(x) \text { can be made as close to } 5 \text { as we like by taking } x$ sufficiently close to 2 (but $x \neq 2$ ).] Yes, the graph could have a hole at (2,5) and be defined such that $f(2)=3$.

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Video Transcript

Okay, what is meant by the equation? The limit as X approaches to equals five. So let's just talk about this. The limit as X approaches to of this function, whatever it is, F of X is equal to five. Now, this part here tells me that as X approaches to from the right and from the left, ah that should be a minus there. Sorry about that, that the Y value is equal to five. Let's let's draw this and see what it looks like. This function F of X. Very twirly wahoo. But as X approaches to the X axis, the y value is going to be equal to five on both the right and the left. All this means is that from the right and from the left, The function approaches five at The function equals five, not approaches equals five at x equals two. These are both headed towards the point. Y equals five at 2. So what about the other part, is it possible for this to be true? And yet F of two equal to three? The answer is yes. That's totally possible. There could be what we call a hole here Where F. 0. 2 is actually equal to three. And this is still a continuous function still keeps going. The limit as X approaches to is equal to five, but the actual value is just equal to three. So just think of it as like a little hole in the function. They can think of it as a piece wise. So for F of X, it's just yeah, So when x equals two F of two. Well let's just yeah, let's just keep going with that F. F two, it goes three. But for the rest of the function it's just F. Of X. Think of it that way. So it's just a regular function. But at one there's just like a transposed little point there. So yes, it's possible for both of these to be true, because what the first one is saying is that the function is approaching five, The Y Value five on both sides. The second part is saying the actual value is actually three, but it doesn't matter. These are saying different things and they can coexist.