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Problem 55 Hard Difficulty
(a) Use numerical and graphical evidence to guess the value of the limit
$$ \lim_{x \to 1}\frac{x^3 - 1}{\sqrt{x} - 1} $$
(b) 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?
Answer
(a) Let $y=\frac{x^{3}-1}{\sqrt{x}-1}$
From the table and the graph, we guess that the limit of $y$ as $x$ approaches 1 is 6 .
$$\begin{array}{|l|c|}
\hline x & y \\
\hline 0.99 & 5.92531 \\
0.999 & 5.99250 \\
0.9999 & 5.99925 \\
1.01 & 6.07531 \\
1.001 & 6.00750 \\
1.0001 & 6.00075 \\
\hline
\end{array}$$
(b) We need to have $5.5<\frac{x^{3}-1}{\sqrt{x}-1}<6.5 .$ From the graph we obtain the approximate points of intersection $P(0.9314,5.5)$ and $Q(1.0649,6.5) .$ Now $1-0.9314=0.0686$ and $1.0649-1=0.0649,$ so by requiring that $x$ be within 0.0649 of 1
we ensure that $y$ is within 0.5 of $6 .$
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Video Transcript
So in this problem were asked to use numerical and graphical methods to show. But the limit as x goes to one Of this function, x cubed -1 Over the skirt of X -1 equals. So first of all, numerically let's make a table here of excess and f of X. Okay, we'll have our access go to one first from the left. So if you put .5 into this formula you get 2.987 0.75 in here you get 4.315 0.90. In here You have 5.281 inching our way towards towards one right, 0.93 Gives me 5.49 0.99 Gives me 5.9- five. 0.999. Give Me 5.99- five. All right, so this the jess let the limit As X approaches one from the left of our function Is six, doesn't it? All right. Now Let's go from the other way. Let's start at 1.1 At 6.782 and 1.65 At 6.501 1.01 Is 6.075. 1.0 No one said get closer and closer to one. 6.075. So this suggests that the limit As X approaches one from the right then right, I'm coming in closer from the right Of our function is also six. So since I have the same limit from the left and the right then I have the limit. It's not writing very neatly. Let me try that again Limit as X approaches one of x cubed -1 over The Square Root of X -1. This function is six numerically, isn't it? All right. Let's see it graphically. Now I look at it graphically, here's the graph Y equals execute -1 over the square root of X -1. And what do I see as I get closer and closer and closer to one here I can't have the value at one but I can't get closer and closer and you notice that the values are getting closer and closer and closer. 26 all the time. Okay so that means graphically we were able to confirm this as well. Now the next question we're asked is how close 21 his ex for us to be within 0.5 of a limit. Well let's look at the table we made here. Okay The limit is six then I want to be at 5.5 to 6.5 don't I? And I can see right there that I have to be at .93 Or 1.065. Right Okay so I have to be 0.93 So that's equal to X is less than or equal to 1.065. Right within that range. And so I'm only allowed to give one delta here. So you will say that delta is absolute value X -1 for this. Um, where This limit as x approaches one Execute -1 over the square root of X -1. All right, that was the value of this um, -1, modest the limit. Right is within 0.5 Annex. I shouldn't say equal to is within their All right. So for that to happen, that means X is in this range, I can only get one value of delta So that delta is going to be, it's either .07 or .065 so I need the smaller one 0.65 is the answer that I'm looking for.
Oklahoma State University
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