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Use the definition of continuity and the properties of limits to show that the function is continuous at the given number $ a $.

$ f(x) = (x + 2x^3)^4, \hspace{5mm} a = -1 $

\[\lim _{x \rightarrow-1} f(x)=\lim _{x \rightarrow-1}\left(x+2 x^{3}\right)^{4}=\left(\lim _{x \rightarrow-1} x+2 \lim _{x \rightarrow-1} x^{3}\right)^{4}=\left[-1+2(-1)^{3}\right]^{4}=(-3)^{4}=81=f(-1)\]

By the definition of continuity, $f$ is continuous at $a=-1$

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

University of Washington