Graphical limit at point discontinuity

So here we have the graph ( y = G(x) ). We have a little point discontinuity right over here at ( x = 7 ), and what we want to do is figure out what is the limit of ( G(x) ) as ( x ) approaches 7.

So essentially, we say, "Well, what is the function approaching as the inputs in the function are approaching 7?" Let's see. If we input as the input to the function approaches 7 from values less than 7, so if ( x = 3 ), ( G(3) ) is here. ( G(3) ) is right there. ( G(4) ) is right there. ( G(5) ) is right there.

( G(6) ) looks like it's a little bit more than or a little bit less than -1. ( G(6.5) ) looks like it's around -1/2. ( G(6.9) ) is right over there; it looks like it's a little bit less than 0. ( G(6.999) ) looks like it's still less than 0; it's a little bit closer to 0. So it looks like we're getting closer as ( x ) gets closer and closer, but not quite at 7. It looks like the value of our function is approaching 0.

Let's see if that's also true from values for ( x ) greater than 7. So ( G(9) ) is up here; it looks like it's around 6. ( G(8) ) looks like it's a little bit more than 2. ( G(7.5) ) looks like it's a little bit more than 1. ( G(7.1) ) looks like it's a little bit more than 0.

( G(7.1) ) looks like it's a little bit more than 0. ( G(7.01) ) is even closer to zero. ( G(7.00000001) ) will be even closer to zero. So once again, it looks like we are approaching zero as ( x ) approaches 7, in this case as we approach from larger values than 7.

This is interesting because the limit as ( x ) approaches 7 of ( G(x) ) is different than the function's actual value ( G(7) ). When we actually input 7 into the function, we can see the graph tells us that the value of the function is equal to 3. So we actually have this point discontinuity, sometimes called a removable discontinuity, right over here.

I'm not going to go into a lot of depth here, but this is starting to touch on how 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 at that point. Well then we're probably talking about, or actually we are talking about, a discontinuity.