ShmoopTube

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Here's an unshmoopy question you can write about tonight in your diary...

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Which of the following are true about the pictured function?

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I.   This function is not continuous at the point a.

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II.  This function doesn't have a limit at point a.

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III. This function isn't defined at point a.

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Here are the potential answers...

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OK, so we're given a graph and being tested on a few vocab words:

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"Continuous"... "limit"... and "defined."

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No short cuts here so we need to brute force and test each answer.

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One... is the function continuous at point A?

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Well, continuous means the function is defined at point A, and there's no hole in it.

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From that hole and the actual point A, there's a pretty big jump...

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...in other words, the function is NOT continuous at point A.

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This is practically the textbook definition of DIScontinuous.

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Ok check! One is true. So we can eliminate B, C, and E.

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Movin' on.

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Two... does the function have a limit at point A?

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Well, to find the limit at point A, the right and left limits of point A must exist... that

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is, they aren't divided by zero... and have the same values.

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But if we look at the graph, the right and left limits of point A clearly have different values.

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...here... and here...

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The values are some positive y-value and a negative y-value, so... different.

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The function does NOT have a limit at point A. 2 is true, so check!

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We don't even have to check number 3 because there's no I, II, and III option. So D's our answer.

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But for you overachievers, let's take a look at 3.

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Is the function defined at point A?

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Well, yes it is; there's a filled point at point A. So 3 isn't true.

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So D is our answer...done.

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