00:01
Hi everyone, in this video i'm going to show you how to find the limit as x goes to 0 of 1 plus x minus e to the x all over 4x squared using a known taylor series for e to the x, which i have written here below.
00:14
So i'm going to replace e to the x in our function we're taking the limit of.
00:19
I'm going to replace that with the tailor series.
00:21
Okay? so limit as x goes to 0, sorry, of 1 plus.
00:32
X and now we're going to replace e to the x with our taylor series keeping in mind that each term of the taylor series is going to be multiplied by that negative so 1 plus x minus x squared over 2 factorial minus x cubed over 3 factorial minus x to the 4 over 4 factorial and so on all over 4 x squared and we see things cancel out, 1 minus 1, x minus x, sorry.
01:13
So we're left with the limit as x goes to 0.
01:19
Again, i need to see that, there we go.
01:22
Okay, limit as x goes to 0 of a negative x squared over 2 factorial, minus x cubed over 3 factorial, minus x to the 4, over 4 factorial, all over 4x squared.
01:39
Okay, now we can divide each term individually by 4x squared, and simultaneously i want to work out what these factorials are.
01:50
So limit as x goes to 0 of negative x squared over 2 factorial is just 2 times 4 is 8 x squared in the denominator, minus x cubed over 3 factorial is 6 times 4 is 24 x squared minus x to the 4 over 4 factorial which is 24 times 4 is 96 and so on so these x squareds are going to cancel that one will cancel and leave just 1 in the numerator there those will cancel leaving 2 in the numerator there and so on, limit as x goes to 0 of negative 1 8th, minus x over 24, minus x squared over 96, and so on...