Determine the end behavior of the following transcendental fun...

Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.

$f (x) = 1 - \ln x$

Accepted Answer

Hello there, today we're gonna solve the following practice problem together. So first stop, let us read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. Consider the transcendental function F of X equals five minus two, multiplied by the natural log of X, determine the end behavior of this function by analyzing the appropriate limits. Sketch a graph of the function showing asymptotes if they exist. Awesome. So it appears for this particular problem we're asked to solve for two separate things. Firstly, we're trying to determine the end behavior of the specific function by analyzing the appropriate limits. And we're also asked to sketch a graph of the function showing asymptotes if they exist. Awesome. So now that we know that we're solving for two separate things. So our first, once again, we're trying to figure out the end behavior. Firstly, and then secondly, we're trying to sketch a graph showing the asymptotes that they have any. And we're trying to figure out what the sketch of the graph of this specific function will be awesome. So with that in mind, let us quickly first off recall and note that the natural logarithm function alan of X is undefined or so let's see. So undefined, so undefined. So it's undefined at once again, you need to recall that the natural log of X will be undefined at X is less than or equal to zero. So our first step is we need to analyze the limit as X approaches zero from the right. So let's put that what we said in words into writing. Awesome. So with that said, we need to take the limit as X approaches zero from the right, because the plus sign in the power, so the power to the plus sign. So the power of this like addition symbol means that we are analyzing zero from the right. So now that we are analyzing the limit as X approaches zero from the right as approaches zero from the right. So we need to figure out that the natural log of X as the natural log of X approaches negative infinity as X approaches zero from the right. So the natural logarithm of X, so Allen of X approaches negative infinity and X approaches zero from the right. So X approaches zero from the right, which I'm gonna put a capital R from right, for the denote right. So therefore considering these two factors, we can go ahead and write that the limit as X approaches zero from the right of F of X will be equal to the limit as X approaches zero from the right of five minus two multiplied by the natural log of X which is all equal to five minus two, multiplied by the natural log of zero from the right, which is equal to five minus two, multiplied by negative infinity. Awesome. And this will be all equal to five plus infinity, which is all equal to infinity. So this will mean that as X approaches zero from the right F of X will increase without any bound, so it will increase without a bound. So as X approaches zero from the right, we'll bring back our capital R for right, that will mean that F of X will increase without bond. So I'll put an arrow going upwards to show that it's increasing without a bound. Awesome. So with our next step now is we need to analyze the limit as X approaches infinity. So as X approaches infinity, we need to note that as X approaches Ln N, so as X approaches infinity, Ln of X will also approach zero. So Allen of X will also approach infinity. So, but it's important to note, let's take a moment here to pause. So as we should recall a note since in this case, Ln FX will also approach infinity but at a slower rate. So therefore, we can go ahead and write that the limit as X approaches infinity of F of X is gonna be equal to the limit as X approaches infinity of five minus two, multiplied by the natural log of X is equal to five minus two, multiplied by the natural log of infinity. Awesome. And then we're gonna say that this is equal to drum roll, five minus two multiplied by infinity. Which will mean that this is all equal to five minus infinity, which will mean that this is all equal to negative infinity. So therefore, this will mean without a doubt that as X increases without bound F of X will decrease, I'm gonna do a downwards arrow. So to represent decreasing, so F of X will decrease without a bound. So to quickly say that again. So this means since we got an answer of negative infinity, this will mean that X will increase without a bound and F of X will decrease without a bound. Awesome. So now our next step is we need to identify the asymptotes if there are any asymptotes. So based off of our analysis that we made together the function will have a vertical Asymptote at X equals zero. So let's make a note of this. So I'll call this va so vertical Asymptote and the vertical Asymptote will be at X equals zero. And this is because or due to the fact that since the function is not defined for X is less than or equal to the zero and will approach infinity. As X approaches zero from the right, there will be no horizontal Asymptote. So we'll say H A for horizontal Asymptote. And we'll say that this equals none. None, none, none. OK. So there is no horizontal Asymptote but the function will decrease without bound as X increases. So now we can safely based off of the information we have put together, we can officially sketch our graph. So I've gone ahead and put together a fancy graph that I got for my graphing calculator. So you can use a graphing calculator or any graphing software that makes sense to you and it's easy to use. But when we plug in our function into our graphing software, we will receive the following graphs. So let me sketch it. So as we can see, we have a red dotted line going acro going along the vertical axis, the Y axis and that denotes our vertical Asymptote, which is X equals zero. So that is our red dotted line going in vertically. And as you can see, we have no horizontal Asymptote and we could see that our graph which is like L shaped will intersect our X axis, the horizontal axis at about 11 or 12, it appears on our graph from our sketch. So as you could see when we draw our vertical Asymptote, which is along the Y axis, we have a red dotted line and that will be once again at X equals zero. So I'll make a little note here and read off to the side that vertical Asymptote is at X equals zero. We now need to mark that the function is increasing without bound as X approaches zero from the right, as we could see in our graph that I provided. And, and we should also be able to recall and note that as X increases the function decreases without bound, which indicates this downwards trend. As we could see, our graph is going from like our 0.20. It appears on a graph on the Y axis and it goes down and it dips below the horizontal axis. So it's gonna go and it's gonna meet the X AX C and it'll intersect about like 11 or 12 and then it will dip below the X axe and then go onwards. So it's indicating this downwards trend. And lastly, as we should recall a note that the function is undefined that X is less than or equal to zero. So only sketch the graph for where X is greater than zero. So that's what our sketch is indicating. So this is, I'll make a note here in blue. So this is for where X is greater than zero. So in the end, as we should recall, a note, our sketch will show a curve that starts very hard, like very high. So as we could see our sketch shows a curve that starts very high near the vertical Asymptote, which once again is X equals zero and will begin to decrease without bound as X increases. And it will always stay to the right of the Y axe, as we can see in our sketch that I provided to us. And that is it that is we've officially solved for this problem. So to quickly recap here. So for our once again, because we were told by the prom itself to figure out what the, and we were trying to figure out what the N behavior was. So let's write that down. So we're told once again, from what we discovered together that the limit as X approaches zero from the right of five minus two, multiplied by the natural log of X will be equal to infinity. And the limit as X approaches as X approaches infinity of five minus two multiplied by the natural log of X will be equal to negative infinity. And that is our end behavior once again to quickly recap. And that's it we've officially solved for this problem. Hooray. So let's go look at our multiple. Well, there is no multiple choice either. So let's look at our problem itself and this triple check that we solve for everything. So we determined our end behavior which once again to quickly recap again. So our end behavior was positive infinity and negative infinity for our two separate limits. And we created a sketch. So that means we've solved for all parts of this problem and that's it. Thank you so much for watching. Hopefully, that helped. And I can't wait to see you in the next video. Bye.

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.
$f (x) = 1 - \ln x$

Key Concepts

Limits

Transcendental Functions

Asymptotes

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