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# Find the limit, if it exists. If the limit does not exist, explain why.$\displaystyle \lim_{x \to 3}\left( 2x + | x - 3 | \right)$

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##### Kristen K.

University of Michigan - Ann Arbor

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to evaluate this limit, note that The absolute value of X -3 has two values. The first one will be Negative of X -3 If X -3 is less than zero Or that access to less than three And it will be a positive X -3. If X -3 is greater than zero or That X is greater than three. And so to find this limit, you have to do the one sided limits first. That means we do limit as X approaches three from the left of two, X plus. In this case we will use Negative of X -3. Since you're looking at Left values of three and evaluating we will get two times three minus we have three minus three. This gives us a six and the limit As X approaches three from the right, We will then use X -3 for the absolute value and so we have two X plus x minus three and this will give us two times three Plus 3 -3 Which is also equal to six. Now, since the one sided limits are equal, let me say that the Limit as X approaches three of to express the absolute value of X -3 exists and it is equal to six. That means we have limits As X approaches three of two, X plus Absolute value of X -3, this is equal to six

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