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(a) Use a graph to find a number $ \delta $ such thatif $ 2 < x < 2 + \delta $ then $ \dfrac{1}{\ln (x - 1)} > 100 $

(b) What limit does part (a) suggest is true?

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05:50

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 4

The Precise Definition of a Limit

Limits

Derivatives

Campbell University

Harvey Mudd College

Baylor 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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Alright, here's a fun little problem. We have our function F. Of X which is one over Ln X minus one. You can see we have a vertical ascent toad at two because that's what makes um the bottom go to zero because we get L. N. F. One which is zero. And we're going to have a vertical ascent over there. We also know the argument of Ln can't go negative so we can't go um we can't use X equals one or um uh because Ln of zero does not exist because what kind of approach is minus infinity. And so we don't want that point at one. And nor can we go to the left because we can't have a negative argument for Ellen anyway, so that's kind of a fun Uh kind of graph for um for this function whatever Ln X -1. Alright, so our goal was, if we want F. Of X to be greater than 100 then what should delta B. In other words, um Clearly, if I'm at two, I'm gonna be at infinity war approaching infinity. Right? Because I'm at the vertical ascent to it. So I'm way, way, way up there. So in other words, how far out can I go to the right of two And still be over 100. So that's what we're looking for. And so basically what you can do is use your calculator and really zoom in in this location, you have to get really close to find the intersection. So if I find the intersection we find out that the x value for the intersection is 2.01 005. And so at that point, if X is 2.1005, we actually get at that point we get one over Ln um well I'll go ahead and put the number in. That number is equal to 100. Okay, so we want if we go any further out any further to the right Then we're going to be less than 100. Right? So, so basically the idea is that if X has to be in between two and 2 plus delta, Then this part is our delta. So our delta then is just zero 01005. And we can't go any further out from two than that To stay over 100. So um okay, so that is how you find the delta. And then just you can tell from the graph. But basically, if you look at the limit Limit as X approaches two from the right side, we can see that our function is approaching positive infinity. So that's plus infinity. Because it has a vertical accident going up. And you can also notice that on the other side, if we were to approach to the minus side, then we would be uh approaching minus infinity. So anyway, hopefully that helped a nice little look at this funny function. Okay, alright. Take care and have a wonderful day

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