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Find the limit, if it exists. If the limit does not exist, explain why.

$ \displaystyle \lim_{x \to 0^-}\left(\frac{1}{x} - \frac{1}{|x|} \right) $

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$\lim _{x \rightarrow 0^{-}} \frac{2}{x}$ which doesn't exist since denominator approaches 0 and numerator doesn't. To clarify this limit would be equal to negative infinity, but sinceinfinity is not a limit it would therefore be considered as not existent i.e. D.N.E.

02:03

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

Leo L.

January 9, 2022

1/x - 1/(-x) is not 1/2x. Is it not 1/x + 1/x and is equal to 2/x. And then x is not zero. It is approaching from the left. So why not negative infinity? When do you say Limit does not exist and when do you say negative or positive infinity?

Campbell University

Harvey Mudd College

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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Alright, so here we have a fun limit problem. We have limit as X goes to zero from the left side of one over X -1 over absolute value of X. Well the trick is that if I'm approaching zero from the left side then my number is always going to be negative which means the absolute value is going to change the sign. So I can rewrite this as X goes to zero from the left side of one over X minus one over minus X because that absolute value again will change a sign. So therefore I can rewrite this as X goes to zero minus of one over X, I've got a double negative so I can add the second then one over X. Um so keep going down here. So that's the limit As X goes to 0- of two over X. Well if I were to graph that so you can get an idea of what's happening, We know one over X is your basic curve like this and two over X is just a stretch. It's just stretched vertically so the overall same shape. So if I approach zero from the left side I'm going to be going towards minus infinity. So therefore this answer is minus infinity. Or you can always write it as does not exist because infinity or minus infinity does not exist. We never get there but fun to analyze our problems. So okay, have a great day. See you next time

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