WEBVTT
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defined in three different parts. Uh function is defined
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to be one plus X. When x is less
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than negative one, that's the red portion of the
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graph that you see. The function f of X
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equals X squared. When uh negative one is less
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than or equal to ex uh ex in turn is
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strictly less than one. That's the blue portion of
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the graph that you see. And last but not
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least when x is greater than or equal to one
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. Uh F of X is defined to be two
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minus X. We want to find all values a
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along the X axis for which the limit exists.
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And this graph is uh going to greatly help us
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. You can see um that this uh function is
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basically connected. There's no breaks anywhere in the function
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except right here. When X is negative one,
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we have a break into function when X is positive
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one. Uh You can see the blue portion of
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the function uh meets nicely uh the green portion of
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the function. And so the limit as X approaches
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one from the negative side, the blue side is
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going to be the same limit as X approaches one
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from the uh positive side. The green side.
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But over here, it when uh X is negative
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one, or specifically when a is equal to negative
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one. Uh This function uh does not have a
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limit in other words. Uh The limit of f
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of X as X approaches negative one does not exist
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. Um The limits exist for dysfunction for any other
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value, any other real number value along the X
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axis, it does not exist. The limit of
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dysfunction does not exist when X approaches the value of
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negative one because as X approaches negative one from the
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negative side to function uh approaches zero. So the
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limit of F of X as X approaches negative one
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from the negative side is zero, but the limit
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of F of X pay attention to the blue portion
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of the curve. Now the limit of F of
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X as X approaches negative one from the positive side
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, uh F of X is approaching the value of
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one. And so the limit of F of X
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as X approaches negative one does not exist because the
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limit of F of X as X approaches negative one
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from the negative side does not equal the limit of
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F. Of X as X approaches negative one from
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the positive side. And you can clearly see that
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because the red portion of the graph is getting close
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to zero, think of the Y axis as to
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function values, whereas the blue portion of the graph
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is getting close to the value of one