Problem 5 Easy Difficulty

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

$ \displaystyle \lim_{x \to 0} f(x) = -\infty $, $ \displaystyle \lim_{x \to -\infty} f(x) = 5 $, $ \displaystyle \lim_{x \to \infty} f(x) = -5 $

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01:54

Daniel J.

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Video Transcript

Okay, in this problem we want to sketch a function F of x. Uh that satisfies the three given conditions in the problem. Uh First the limit as X approaches zero of our function is negative infinity. So as we approach zero from the right side and from the left side, uh the function is going down towards negative infinity. Okay, so Near 0 to function needs to be going down towards negative infinity from both sides of zero. But before we draw that in uh I want to look at a couple of restrictions. Uh the other condition, the limit of the function as X moves towards negative infinity is five. So his ex moves towards negative infinity. The function goes near five. So his ex is moving to the right, The functions going near five. So let's draw that little portion it as X moves to the left towards negative infinity to function goes towards five. So here you have as X is approaching negative infinity to function is approaching the value of five. Um as X approaches zero, the limit of the function as X approaches zero was negative infinity. So as we approach zero, as X approaches zero to function needs to move down towards negative infinity. So as we get closer to zero, X equals zero to function goes down towards negative infinity. Okay, so let's continue this curve here. So his ex moves closer to zero to function is heading down towards negative infinity. Now, don't forget, as X approaches uh zero Fund a positive side. Uh the function still has to move down towards negative infinity. And that's because if the limit as X approaches zero of our functions negative infinity. Uh That means uh to function has to go down towards negative infinity as we approach zero from both sides from the left side of zero and from the right side of zero. But before we start drawing this, let's look at our third condition. The third condition says the limit of our function as X moves towards positive infinity is negative five. The limit of F of X as X approaches infinity is negative five. So as X approaches positive Phineas, ex moves to the right, The function approaches -5. But don't forget as X approaches zero, the function has to go towards negative infinity. So as we're getting close to zero, the functions going down towards negative infinity. And as X is moving to the right, the functions going towards uh negative five. So we want something that kind of is going to get drawn in like this celeste draw that as we get close to zero to function is going to be dropping down towards negative infinity. And as Ex moves towards the right towards positive and funny, the function is going to get close to negative fuck. Alright. So here this blue graph is the graph of a function F of X that satisfies the conditions as ex moves towards negative infinity to limit of a function will be five. As ex moves towards zero. The limited function is negative infinity, it's dropping down steeper and steeper and steeper. As ex moves towards positive infinity. To function approaches -5.