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(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?
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04:22
Daniel Jaimes
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 2
The Limit of a Function
Limits
Derivatives
Campbell University
Oregon State University
University of Michigan - Ann Arbor
Idaho State University
Lectures
04:40
In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
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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.
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