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Prove that $ \displaystyle \lim_{x \to 0^+} \ln x = -\infty $.

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 4

The Precise Definition of a Limit

Limits

Derivatives

Baylor University

University of Nottingham

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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This is a problem. Number 43 of the Stuart Calculus ES edition section 2.4 prove that the limit has experts zero from the right of the function Ln of X is equal to negative infinity, and we're going to use a little bit of definition seven and definition for definition, for states for red hand limits and that there should exist adults a value such that between a X or between a and A plus steel toe. The function has a certain conditional, but in this case, this is an infinite limit. So from definition seven, what we have is that we want to choose a value, and that is positive that if this is true, then this function is always going to be less than M. So this is a bit of a combination between definitions four and seven. So we're gonna do is we're going to take first. We're going to understand what this is in our case, A zero since we're approaching zero from the right. Therefore, we're in this area between X zero and delta the function it is olympics less than in. And if we wanted to choose an appropriate ex, we take maybe the solve for X. We'd take the exponential to both sides. And by comparing these two cases here, we see that an appropriate choice of Delta in order to be consistent with this definition of a limit is e to the power capital in so the states that for any end, we are able to find a delta that is positive and is consistent with these conditions, Therefore, proving that this limit is equal to negative infinity.

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