now let's talk about how to find the value of a limit from a graph let's just make some points first so let's say if we have an open circle at this point a closed circle and also another open circle what is the limit as x approaches 2 from the left side so this is known as a one-sided limit and let's call this function f of x so basically this question is asking you as x approaches 2 from the left what is the y value so here is an x value of 2 as we approach it from the left side notice that the y value of that curve is three this is one two three now what is the limit as x approaches two from the right side of f of x now if we follow the curve from the right side as we approach an x value of 2 notice that the y value is negative 4. now how about the limit as x approaches 2 from either side because the left side and the right side do not match this limit does not exist now what about the value of f of 2 when x is exactly 2 what is the y value so we have to look at the closed circle the y value is 1 so f of 2 is equal to 1. now let's try another example let's say if we have an open circle at that point and a closed circle here what is the limit as x approaches positive 3 from the left side and this is the graph of f of x so here is an x value of three as we approach it from the left side notice that the y value is two so this is equal to two now what is the limit as x approaches three from the right side so as we approach an x value of 3 from the right side the y value is still 2. now what is the limit as x approaches 3 from either side because these two values are the same the limit exists it's equal to 2. finally what is the value of f of 3 when x is exactly three what is the value of y so whenever you get a question like that look for the closed circle the closed circle has a y value of four so f of three is equal to four consider this one so what is the limit as x approaches 2 from the left side and what is the limit as x approaches 2 from the right side and then find the limit as x approaches 2 from either side and in addition to that find the value of f of 2. so go ahead and pause the video go ahead and find the answers to these problems so let's start with the first one so here's an x value of two as we approach that point from the left side notice that the y value is also two now as we approach x equals two from the right side the y value is still two so therefore the limit as x approaches two from either side must be two as well because the left side and the right side have the same y value now what is the value of f of 2 so when x is exactly 2 what is the y value notice that there's no closed circle in this particular example so therefore it does not exist if you don't see a closed circle at an x equals two f of two is not defined now let's look at a fourth example what is the limit as x approaches 4 from the left side and also find the limit as x approaches 4 from the right side and as x approaches 4 from either side and also find the value of f 4. so what is the limit as x approaches 4 from the left looking at the graph here's an x value of four so as we approach from the left side notice that the y value is approximately negative two now as we approach positive four from the right side the y value is still negative two so the limit exists since the left side is the same as the right side the limit as x approaches four from either side is also negative two now what about f of four notice that we do have a closed circle at x equals four so it's also equal to negative two so therefore everything is the same the function is continuous at an x value of four here's another problem you can try so what is the limit as x approaches 3 from the left side let's start with that so what do you think it's equal to so here's three it's the vertical asymptote and as we follow the curve as we approach three from the left side notice that it doesn't converge to a specific y value it keeps getting lower and lower and lower so therefore it goes to negative infinity now what about the limit as x approaches 3 from the right side if we follow the curve towards the vertical asymptote with an x value of 3 notice that it keeps going higher and higher and higher so therefore it approaches positive infinity now what is the limit as x approaches 3 from either side because these two do not match the limit does not exist now what about f of 3 we don't have any closed circle at x equals 3 so therefore f 3 is not defined it doesn't exist let's try another similar example what is the limit as x approaches positive 4 from the left side given the graph of f of x so here is the x value of four and as we approach it notice that the curve increases it keeps getting higher and higher so the y value approaches infinity that is positive infinity now as we analyze the limit as x approaches 4 from the right side as we follow the curve towards positive 4 from the right it also keeps going higher and higher it goes towards positive infinity so therefore what is the limit as x approaches 4 from either side since these two are the same you can see that it approaches positive infinity now granted infinity is not really a number i mean you infinity could be a million a billion a trillion or just something that just it's never ending it's just keep increasing so because it does a conversion specific value you could also say that this limit does not exist some textbooks will have this answer some will have infinity so infinity really doesn't exist it's just a way of describing what happens as x approaches four that is the curve gets bigger and bigger and bigger the y value that i say it gets higher and higher and higher and we know that f of 4 is undefined it doesn't exist we don't have a closed circle at x equals 4. now let's work on some examples with horizontal asymptotes so let's say if we have a horizontal asymptote at y equals three and another one at y equals negative four and let's say the first graph well let's say we have a curve that looks like this actually what is the limit as x approaches positive infinity what is the answer so this is saying when x gets very large that means as we follow the curve all the way to the right and what is the y value notice that it approaches the horizontal asymptote so it has a y value of 3. now what about the limit as x approaches negative infinity so if we follow the curve all the way to the right notice that the y value that it approaches is negative 4. so that's how you can evaluate limits at infinity when x becomes very large you need to find the y value of the horizontal asymptote when y becomes very large when y approaches infinity you need to look for the vertical asymptote