00:01
So we want to find the slope of the tangent line of the function f of x is equal to two times the square root of x at the point 4 comma 4 using our limit definition.
00:09
Our limit definition being the limit as h approaches zero of our modified function f of x plus h minus our original function f of x over h.
00:17
I do suggest to have that formula memorize as it is important to know.
00:22
So when we start, we're going to have the limit as h approaches zero of our modified function.
00:28
So 2 times the square root of x plus h minus our original function 2 times the square root of x over h.
00:40
Now in order to simplify this to something we can work with, we're going to multiply by the conjugate, which is our numerator 2 times the square root of x plus h, and instead of doing minus 2 times the square root of x for an energy plus 2 times the square root of x.
00:57
And this is so that this can be simplified.
01:00
And since we're going to multiply by this on the top, we also have to multiply this by the same thing on the bottom.
01:12
So our next step will be able to foil this, multiply our firsts, our insides, and our lasts.
01:23
And when we foil it, we're going to end up with the limit as age approaches zero.
01:32
And we're going to do our outsides.
01:34
So we have 4x.
01:39
Or sorry, we're going to do our firsts.
01:41
So we're going to have 4x plus.
01:44
Plus 4h because our square roots we're going to cancel out then we're going to do our two times our square root of x plus h times two times the square root of x so we're going to have plus four times the square root of x plus h times the square root of x now we're going to do negative 2x times the square root of x plus h so we're going to have negative four square root of x then we do our last, which they'll give us negative 4x...