Use numerical and graphical evidence to guess the value of...

Use numerical and graphical evidence to guess the value of the limit lim_(x->1)(x^(3)-1)/(sqrt(x)-1) How close to 1 does x have to be to ensure that the function in part (a) is within a distance 0.5 of its limit? Use numerical and graphical evidence to guess the value of the limit x3-1 lim 1x-1 SHOW ANSWER How close to 1 does x have to be to ensure that the function in parta is within a distance 0.5of its limit?

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00:01 So in this problem, we're asked to use numerical and graphical methods to show what the limit is x goes to 1 of this function, x cubed minus 1 over the square root of x minus 1 equals.

00:30 So first of all, numerically, let's make a table here of x's and f of x.

00:41 Okay.

00:43 We'll have our xs go to 1, first from the left.

00:46 So if you put 0 .5 into this formula, you get 2 .987, 0 .75 in here.

00:56 You get 4 .315.

01:00 0 .90 in here.

01:02 You get 5 .281.

01:07 Inching our way towards 1, right? 0 .93 gives me 5 .49 .0 .99.

01:21 Gives me 5 .925.

01:26 0 .999.

01:30 Gives me 5 .9925.

01:33 All right.

01:36 So this suggests that the limit as x approaches 1 from the left.

01:42 Of our function is six doesn't it all right now let's go from the other way let's start at 1 .1 that's 6 .782 and 1 .065 is 6 .501 1 .01 is 6 .075 1 .01 is 6 .075 1 .001 see i'm getting close to closer to 1, 6 .075.

02:27 So this suggests that the limit as x approaches 1 from the right then, right? i'm coming in closer from the right of our function is also 6.

02:40 So, since i have the same limit from the left and the right, then i have the limit.

02:49 It's not writing very neatly.

02:50 Let me try that again...

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