here we have a piecewise function and we're asking if it's continuous at x equals negative one and the function says when X is not equal to negative one we have x squared plus X over X plus one and is simply 2 when X does equal negative one to approach this we use our continuity checklist the continuity checklist gives three requirements for a function to be continuous at a point so here our a is negative one right so let's check through these three requirements is the function defined at negative one so let's see here so this is one all right so f of negative one well I go back up this is a piecewise function and here right here it says what happens at x equals negative one f of negative one is two so yes it is defined there the function is simply two at negative one okay moving on to number two in the continuity checklist here number two says does the limit exist so here we're approaching negative one so I'm going to evaluate the limit as X goes to negative one of this function of f of x but remember that the limit as X approaches negative one doesn't actually care what happens at the point x equals negative one it only cares what happens as we get very close to x equals negative one so I'm going to focus only on this part of the limit here this part of the function and I'm going to ignore what happens to the function at x equals negative one right we just want to know what happens when X is not equal to negative one but getting very close to negative one thus instead of writing f of x here for the function I'm going to write x squared plus X over X Plus 1. well if I just plug in negative 1 to that I'm getting 0 over 0 highly problematic let's see if we can get some things to cancel we can right that numerator factors quite nicely so we have the limit as X goes to negative 1 of x times X plus one and look at that all over X Plus 1 and we get our nice cancellation now we can just plug in x equals negative 1 here so the limit as X goes to negative 1 of X is simply negative one okay so far so good moving on to requirement number three in the continuity checklist we now need the limit as the function approaches negative one so we need the limit as X goes to negative one of f of x to be equal to f of negative one and now we have problems right because the limit we decided was negative one but when we evaluate the function at negative one we decided that was two and those are not equal okay so in conclusion we can write that as three dots like that we conclude that f is not continuous is not continuous okay at that point at x equals negative one this actually continuous at every single other point in the real numbers except for x equals negative one but we are focusing on the one point that this function is not continuous