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Problem 9 Easy Difficulty

Sketch the graph of an example of a function $ f $ that satisfies all of the given conditions.

$ f(0) = 3 $, $ \displaystyle \lim_{x \to 0^-} f(x) = 4 $, $ \displaystyle \lim_{x \to 0^+} f(x) = 2 $, $ \displaystyle \lim_{x \to -\infty} f(x) = -\infty $, $ \displaystyle \lim_{x \to 4^-} f(x) = -\infty $, $ \displaystyle \lim_{x \to 4^+} f(x) = \infty $, $ \displaystyle \lim_{x \to \infty} f(x) = 3 $

Answer

$$\begin{array}{l}
f(0)=3, \quad \lim _{x \rightarrow 0^{-}} f(x)=4 \\
\lim _{x \rightarrow 0^{+}} f(x)=2 \\
\lim _{x \rightarrow-\infty} f(x)=-\infty, \quad \lim _{x \rightarrow 4^{-}} f(x)= \\
-\infty \\
\lim _{x \rightarrow 4^{+}} f(x)=\infty, \quad \lim _{x \rightarrow \infty} f(x)=3
\end{array}$$

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Video Transcript

played a few uh restrictions for starters f of zero equals street. So when x zero D function value is tree, so that's this point right here. Uh The limit of F a bex, as X approaches zero from the negative side is four. So as X approaches zero from negative side to function is going towards four doesn't hit it because at X equals zero to function is here. Um So we're going to be approaching uh the value of for will put an open circle there. Uh As X approaches zero from the positive side, uh F of X is going to approach to so open circle there. Uh the only point on the function when X0 is up here at three. Um as X approaches for from the negative side to function is going to go down towards negative infinity. So we're going to have an ass um Toque line at X equals four. And uh so as X approaches for from the negative side to function goes down towards negative infinity. As X approaches for from the positive side of function is going to go up towards positive Last but not least as x goes towards infinity, Alphabet's approaches three. So let's see if we can draw a function that contains all these situations. As X approaches zero from the negative side to function approaches for sort of function is going to be approaching for As X approaches zero from the negative side and to the left, when extra approaches negative infinity to function uh approaches negative infinity. So we're going to have something that looks like this, Alright, as X approaches zero from the positive side, the function is approaching two. Um But then as X approaches for from the negative side, uh the function goes down towards negative infinity. So it looks like we're going to have something like this and uh we'll check all this one we're done. Uh one item at a time. But now as X approaches four from the positive side to function goes towards infinity. And as X approaches positive infinity, allowing the positive X access to function approaches trade. So approaching four from the positive side, function goes up towards positive infinity. X approaches positive infinity. uh to function approaches three, which is this ascent Opine right here. So I believe this is what our function F of X should look like. Let's read through each restriction and see if it satisfies it. Uh F of zero is three. So when X zero, a point on the graph of F of X right here, at three, as X approaches zero from the negative side to function is approaching for. So as X approaches zero from the left side, the function approaches for open circle because it doesn't hit it. As X approaches zero from the positive side to function approaches to as X approaches negative infinity, the function itself goes down towards negative infinity. As X approaches four from the left from the negative side to function goes down towards negative infinity. As X approaches four from the positive side to functions going up towards positive infinity last, but not least as X approaches positive infinity, to function approaches a value of three.