Limits by factoring

let's say that f of X is equal to x squared x squared plus X minus 6 over X minus 2 and we're curious about what the limit of f of X as X approaches to is equal to now the first attempt that you might want to do right when you see something like this is to see what happens what is f of 2 now this won't always be the limit even if it's defined but it's a good place to start just to see if something reasonable could pop out so looking at it this way if we just evaluate F of 2 on our numerator we're going to get 2 squared plus 2 minus 6 so it's going to be 4 plus 2 which is 6 minus 6 you're going to get 0 in the numerator and you're going to get 0 in the denominator so we don't have the function is not defined so not defined at X is equal to 2 F not defined so there's no simple thing there even if this did evaluate if it was a continuous function then it actually the limit would be whatever the function is but that doesn't necessarily mean the case but we see very clearly the function is not defined here so let's see if we can simplify this and we'll also try to graph it in some way so one thing that might have jumped out at your head is you might want to factor this expression on top so if we want to rewrite this we could rewrite the top expression and this is goes back to your algebra one two numbers whose product as negative six whose sum is positive 3 well that could be positive 3 and negative 2 so this could be X plus 3 times X minus 2 all of that over X minus 2 so as long as X does not equal 2 these two things will cancel out so we could say this is equal to this is equal to X plus 3 X plus 3 for all X's except for X is equal to 2 as long as X does not equal 2 so that's another way of looking at another way we could rewrite our f of X do it in blue just to change the colors we could rewrite f of X this is the exact same function f of X is equal to X plus 3 when X does not does not equal two and we could even say it's undefined it's undefined when X is equal to two so given this definition it becomes much clearer to to us of how we can actually graph f of X so let's try to do it so that is that is not anywhere near being a straight line that is much better so let's call this the y-axis I'll call it y equals f of X and then let's rover here let me make a horizontal line that is my x-axis so defined this way f of X is equal to X plus 3 so if this is one two three we have a y-intercept at three and then the slope is one the slope is one but it's defined and it's defined for all X's except for X is equal to two so this is X is equal to 1 X is equal to 2 so when X is equal to 2 our it is undefined so let me make sure I can so it's undefined right over there it's undefined undefined right right over there so this is what this is what f of X this is what f of X looks like now given this let's try to answer our question what is the limit of f of X as X approaches 2 well we can look at this graphically as X approaches 2 from lower values into so right over it so this is this right over here is X is equal to 2 if we get to maybe let's say this is 1 point 7 we see that our f of X is right over there if we get to 1 point 9 our f of X is right over there so it seems to be it seems to be approaching this value right over there similarly as we approach 2 from from values greater than it if we're at like I don't know this could be like 2.5 2.5 our f of X is right over there if we get even closer to 2 if we get even closer to our f of X is right over there and once again we look like we are approaching this value or another way of thinking about if we ride this line from the positive direction we seem to be approaching this value for f of X if we ride this line from the negative direction from values less than 2 we seem to be approaching this value right over here and this is essentially the value of X plus 3 at X if we said X is equal to 2 so this is essentially going to be this value right over here is equal to 5 if we just look at it visually if we just graphed a line if we just graphed a line with slope 1 with a y-intercept of 3 this value right over here is 5 now we can also try to do this numerically we can also try to do this numerically so let's try to do that so if this is our function definition completely identical to our original definition completely identical to our original definition then we just try values as X gets closer and closer to 2 so let's try values less than 2 so 1.999 this is almost obvious 1 point 9 9 9 9 9 plus 3 plus 3 well that gets you pretty darn close to 5 if I put even more 9 so your got even closer to we'd get even closer to 5 here if we approach 2 from the positive direction and then we once again we're getting closer and closer to 5 from the positive direction if we were even closer to 2 we'd be even closer to 5 so whether we look at it in numerical here we look at it graphically it looks pretty clear that the limit here the limit here is going to be equal to 5