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Evaluate the limit ....

Evaluate the limit ....-example-1
User Lilibeth
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1 Answer

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It depends on how t is approaching 2.

In fact, if you consider the right limit, you have


\displaystyle \lim_(t \to 2^+) (2-t)/(|t-2|)

But since t is greater than 2, t-2 is positive, and thus |t-2| = t-2.

So, the limit becomes


\displaystyle \lim_(t \to 2^+) (2-t)/(t-2) = \lim_(t \to 2^+) -1 = -1

On the other hand, if you consider the left limit, you have


\displaystyle \lim_(t \to 2^-) (2-t)/(|t-2|)

But since t is less than 2, t-2 is negative, and thus |t-2| = -t+2.

So, the limit becomes


\displaystyle \lim_(t \to 2^+) (2-t)/(-t+2) = \lim_(t \to 2^+) 1 = 1

So, this limit does not exist, because the left and right limits exist but are not the same.

User Natalia
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