WEBVTT
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All right. So for this problem we want to
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um use the definition of continuity and the properties of
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limits to show that this function um X plus two
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X cubed all to the power of four is continuous
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at a. Or maybe X equals negative one is
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less confusing to think about. Um Okay so continuity
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. What is the definition of continuity at a?
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We need that the limit. The left hand limit
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must exist exists. We need the right hand limit
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exists. So that's continuity on both sides and we
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need them to be equal and we need them to
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be equal to the value of the function. Okay
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so we have a very very nice function. This
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is a polynomial also. Um That's nice. So
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it's probably gonna be true. So that's good.
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Um Okay so that's basically all we need to check
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um these these are generally going to exist or or
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we'll check that they exist by evaluating them. So
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let's just see the limit as X goes to.
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So we'll put in a is negative one negative one
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from the left of X plus two X cubed plus
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four. There's nothing funny going on here so we
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can just plug in negative one. Sorry my thing
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won't write for a second. Okay so this is
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negative one plus negative one cubes negative ones of minus
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to alter power four. That's negative three to the
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power for I believe that's positive 81 and then um
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F at negative one is gonna be the same thing
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. We're gonna plug in negative one into F.
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So that's also going to equal positive 81. And
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then the limit as X. Goes to negative one
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from the right of dysfunction Same thing. There's nothing
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funny going on so we can just plug it in
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negative 1-2 to the fore Is positive 81.
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So those are all equal. So this thing is
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true. Um So we're done. Um It might
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seem like a lot of unnecessary writing. You just
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plugged in negative one into the function three times.
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But as I said it's because it's a super nice
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function so you didn't have to do much. Um
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If there were some weird thing going on at the
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value you might have to do a little bit more
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. And this definition of continuity actually ends up being
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helpful.