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Current time:0:00Total duration:4:10

so we have an interesting problem or exercise here find a such that the limit as X approaches 0 of the square root of 4 plus X minus the square root of 4 minus a times X all of that over X is equal to 3/4 and like always I encourage you to pause the video and give a go at it so assuming you have had your go now let's do this together so when you just try to superficially evaluate this limit here if X approaches 0 so if you just try to evaluate this one x equals 0 you're going to get so let me just try to evaluate the limit as X approaches 0 of the square root of 4 plus X minus the square root of 4 minus a X all of that over X well this right over here is going to be just the principal root of 4 because 4 plus 0 is 4 this right over here is going to be the principal root of 4 because well no matter what a is a times 0 is going to be 0 so you're going to be left with 4 minus 0 so it's just the principal root of 4 so you're gonna have to this whole thing is going to be 2 if you just were to substitute of X there so this whole thing is 2 this whole thing right over here is going to be 2 as well you're going to have 2 minus 2 and then as exports to 0 this is going to be 0 so this looks like we are going we are getting an indeterminate form and when you get to something like this you start to say well lapa tiles rule might apply if I get 0 for 0 or infinity over infinity well this limit is going to be the same thing as the limit as X approaches 0 this is going to be the same thing as the limit as X approaches 0 of the derivative of the numerator over the derivative of the denominator so what is the derivative of the numerator so actually let's do the derivative of the denominator first because the derivative of X with respect me I'm going to then a different color the derivative of X with respect to X is just going to be 1 but now let me take the derivative of this this business up here the derivative the derivative of this with respect to X so this is 4 plus X to the 1/2 power so this is good the derivative of this part it's going to be 1/2 times 4 plus X to the negative 1/2 power and so the derivative of this part right over here let's see here the the chain rule applied here but the derivative of 4 plus X is just 1 so we just multiply this thing by 1 but here the chain rule the derivative of 4 minus ax with respect to X is negative a now we multiply that I'm going to have this negative out front so this is going to be plus a plus a times times 1/2 times 4 minus a X to the negative 1/2 power I just use the power rule and the chain rule to take the derivative here and so what is this going to be well this is going to be equal to this is going to be equal to something over 1 so we have up here as X approaches 0 this is going to be this part 4 plus 0 is just 4 to the negative 1/2 power well that's just going to be one half for the 1/2 is 2 4 to the negative 1/2 is 1/2 and then as X approaches 0 here this is going to be 4 to the negative 1/2 which is once again 1/2 so what does this simplify to we have 1/2 times 1/2 which is 1/4 that's that there and then over here I have a times 1/2 times 1/2 so that's going to be plus a over 4 and so this is the same thing as just a plus 1 over 4 and we say that this needs to be equal to 3/4 this needs to be equal to 3/4 that was our original problem so that needs to be equal to 3/4 and now it's pretty straightforward to figure out what a needs to be a plus 1 needs to be equal to 3 or a is equal to 2 and we are done