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

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 7 and what we want to do is figure out what is the limit of G of 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 so if we input as the input to the function approaches 7 from values less than 7 so if X is 3 G of 3 is here G of 3 is right there G of 4 is right there G of 5 is right there G of 6 looks like it's a little bit more than or a little bit less than negative 1 G of 6.5 looks like it's around negative 1/2 G of negative 6 point 9 is right over there looks like it's a little bit less than 0 G of negative 6 point 9 9 9 looks like it's a little bit 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 values greater than 7 so G of 9 is up here looks like it's around 6 G of 8 looks like it's a little bit more than 2 G of 7 point 5 looks like it's a little bit more than 1 G of seven point one looks like it's a little bit more than 0 G of 7 point 1 looks like it's a little bit more than 0 G of seven point zero one is even closer to zero G of seven point zero zero zero zero zero zero or one looks will be even close 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 of 7 and this is interesting because the limit as X approaches 7 of G of X is different than the functions actual value G of 7 when we actually input 7 into the function 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 discontent you sometimes called a removable discontinuity right over here and this is I'm not going to 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 at that point well then we're probably talking about or actually we are talking about a discontinuity