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# Find the limit, if it exists. If the limit does not exist, explain why. $\displaystyle \lim_{x \to 0^-}\left(\frac{1}{x} - \frac{1}{|x|} \right)$

## $\lim _{x \rightarrow 0^{-}} \frac{2}{x}$ 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.

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##### Kristen K.

University of Michigan - Ann Arbor

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to evaluate this limit, we know that the absolute value of X this is equal to negative X. If Access less than zero and positive X if X is greater than zero. Now, since excess approaching zero from the left, then we're talking about absolute value of X equals negative X. And so in here we can write this as the limit As X approaches zero from the left of one over X one over negative x. Which you can simplify further into the limit As X approaches zero from the left of one over X plus one over X. Or this is the same as The limited sex approaches zero From the left of two over X Known that as X approaches zero from the left, the value of X is a very small negative number. So The value of two over X would be approaching negative infinity. And so this limit As X approaches zero from the left of two over X must be negative infinity.

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