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Numerade Educator

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Problem 50 Easy Difficulty

Let
$ f(x) = \left\{
\begin{array}{ll}
x^2 + 1 & \mbox{if $ x < 1 $}\\
(x - 2)^2 & \mbox{if $ x \ge 1 $}
\end{array} \right.$

(a) Find $ \displaystyle \lim_{x \to 1^-}f(x) $ and $ \displaystyle \lim_{x \to 1^+}f(x) $.
(b) Does $ \displaystyle \lim_{x \to 1}f(x) $ exist?
(c) Sketch the graph of $ f $.

Answer

a. 2 and 1
b. The limit does not exist because the left and right limits are not equal
c. see the graph

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Video Transcript

this problem Number fifty with the sewer calculus Safe Edition section two point three Let f equal and this is a piece of ice function. So the first function is X squared, plus one if X is less than one and the quantity X minus two squared If X is greater than or equal to one party find the limit is exporters one from the left of F and the limit his expertise one from the right of f the limit as experts is one from the left of we'LL have to do with dysfunction. Since we're approaching one from the left, this only applies to this region. X is less than one, and so our function half will be X squared plus one and through directs an institution we played in one square two plus one gives us our value of two for the first limit for the second limit. The limit, his expressions that one from the right. We are in this region greater than or equal to one, which means that our function is Dequan titty X minus two squared and through direct institution one minutes Tuesday at one. It's quantity squared is positive. One. Prepare B does limit his expertise. One of enough exists. It does not exist because two does not equal one. The left limit is not equal to right. Lim. Therefore, this limit does not exist and finally for part. See, we need This gets aggressive. The function f we need to plot the first function for this region X is less than one and then plot. The second function, X minus two Quantity squared for X is greater than equal to one. Here's an example of ah plotting mechanism where the ranges that don't mean is restricted for each function. Hex QUOTABLES one is a parabola and as we can see as it purchased a positive one, that is where we a counter a hole. And then he jumped down to the next function explains to quantity squared where that continue function continues on after X is equal to one Here. We also see that the limit does not exist. That X equals two because the function approaches a different value two from the left than it does from the right one