5

1 + 2x f() = 3x 2...

Question

1 + 2x f() = 3x 2

1 + 2x f() = 3x 2



Answers

$$f(x)=\frac{1}{x+2}$$

Okay, We're gonna take a look at the function F of X is equal to one plus two eggs raised to the one over X power. And what were interested in is what the function is doing around X equal to zero. So let's go ahead and grab this on a graphing calculator and utilize the features of a graphing calculator. So that's what we do. And I'm a graph one close to X raised toothy, one over X power. Okay. And so we are really interested. And what the graph is doing right here at X, equal to zero. And the grav itself is a little deceiving because it looks like the functions continuous. So let's go ahead and open up the table feature of our calculator and get a better idea. And so we notice right here at X equals zero, my functions undefined. Which means there is a dis continuity point right here. There's a whole right here, X equals zero. And so let's go ahead and get a better idea about what my left side and my right side is approaching. So let's go ahead and change these two smaller incremental values as re approach zero and we're approaching zero from the left, and we're gonna approach zero from the rights. Now, you can do any step you want to kind of like the point. No one steps just to get a quick idea, but you could do 10.1 if you want to. But we notice that as we're approaching zero from the left, the Y values the function values or decreasing. They're going from 7.699 to 7.54 And then if I jump on this other side and I approached zero from the right side, my Y values or my function values or increasing 7.107 point two. So it almost looks like they are going to meet right here at X, equal to zero somewhere, about 7.4 possibly. And so my one side of limits are approaching the exact same value. So let's go ahead and put that back over here. And so we can say the limit as experts zero From the left of my function, the function value approached about 7.4 and the limit as X approaches zero from the right of my function that also approached about 7.4. So since both the one side of limits approached the exact same value the limit of my function as experts zero approaches 7.4, which now we can say that the function evaluated it. Zero can be extended to 7.4, which means the dis continuity point at X equal to zero is removable.

We're gonna use the table to calculate the limit of the function. Negative one over three plus X as X approaches. Negative up. And we can see why. This is why night of three is interesting, because when you plug in negative three for X, the denominator becomes a hero. So but there's a table but x here and f of X on this side. And since again the interesting behavior happens at negative three work into pick numbers that get closer and closer to negative three. What? So here these three choices are all bigger, the negative three. So that means we're calculating the limit approaching from the right. So when I plug in a negative 2.5, I get that F of X is negative, too. When I plug in negative, 2.9 f of X is negative 10. And finally, when you plug in negative 2.99 you get that f of X. It's negative 100. So we see here that a small change in X as we a closer to negative three causes a huge decrease in f of X, which tells us that the right sided limit here is negative infinity Now we're not done yet. We need to calculate the limit coming from the other direction. So the left sided limit. And they're not just putting quotation marks here, so I don't have to rewrite the whole function. And so we're gonna pick similar numbers coming in from the other direction. So let's say negative 3.5 negative. 3.1 and negative. Three point 01 So f of negative 3.5 gives us two f of native 3.1 is 10. Enough of negative. 3.1 is 100. So just as above, we see that a small a small increase in X as we get closer to negative three, cause it a huge increase in f of X, which tells us that this limit is infinity. And you can check this by graphing your function on graphing calculator or well, from Alfa. And if you do, you will see that it looks something like this. James. Sorry I got this. I got this part wrong like this, just as we would expect from our table

Okay in this problem we have F of X equals three X squared minus four X plus one. Uh Have prime of X. Taking the derivative of our function F. A bex the derivative of three X squared, while the derivative of X squared would be two times X to the first. Uh Such derivative of three X squared will be two times three X to the first or six X to the first or 6 6. So two times three X six X. And then we should attract one from that exponent. So it's X to the first, which we don't have to write to do and then bring down the subtraction. Sign. The derivative of four times X is just four and uh the derivative of a constant is zero. So we don't have to write plus zero. So here's F prime of X. Uh So if we want to find what is the derivative of F evaluated when X is a. We simply just plug a into the derivative function. So F prime of X is six times X minus four. F prime of a is six times a minus four.

Functions and we're going to compose them three ways. One is F of G of X, which means the G goes into the F. So we have three all over X plus one squared minus one. So we end up with X squared plus two X plus one minus one. And then that's our final solution. The next one is Gff, so F is going to go into G and I have three all over X squared minus one plus one, and then five goes on the bottom up of five. So we ended up with 3/24 which gives us 1/8.


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