Indlæser

Limits from graphs: limit isn't equal to the function's value

Video udskrift

- [Voiceover] So, here we have the graph y is equal to g of x. We have a little point discontinuity right over here at x is equal to seven, and what we want to do is figure out what is the limit of g of x as x approaches seven. So, since you say, well, what is the function approaching as the inputs in the function are approaching seven? So, let's see, so if we input as the input to the function approaches seven from values less than seven, so if x is three, g of three is here, g of three is right there, g of four is right there, g of five is right there, g of six looks like it's a little bit more than, or a little bit less than negative one, g of 6.5 looks like it's around negative half, g of negative 6.9 is right over there, looks like it's a little bit less than zero, g of negative 6.999 looks like it's a little bit, it's still less than zero but it's a little bit closer to zero so it looks like we're getting closer as x gets closer and closer but not quite at seven, it looks like the value of our function is approaching zero. Let's see if that's also true from values for x values greater than seven, so g of nine is up here, looks like it's around six, g of eight looks like it's a little bit more than two, g of 7.5 looks like it's a little bit more than one, g of 7.1 looks like it's a little bit more than zero, the g of 7.1 looks like it's a little bit more than zero, g of 7.01 is even closer to zero, g of 7.0000001 looks like it'll be even closer to zero so once again it looks like we are approaching zero as x as approaches seven, in this case as we approach from larger values of seven. And this is interesting because the limit as x approaches seven of g of x is different than the function's actual value, g of seven, when we actually input seven into the function. When we actually input seven into the function, we can see the graph tells us that the value of the function is equal to three. So we actually have this point discontinuity, or sometimes called a removable discontinuity, right over here, and this is, I'm not gonna do a lot of depth here, but this is starting to touch on how we, one of the ways that we can actually test for continuity is if the limit as we approach a value is not the same as the actual value of the function of that point, well, then we're probably talking about, or actually we are talking about, a discontinuity.