Horizontal and vertical asymptotes a. Evaluate limx →+∞ f(x)...

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the variable approaches positive or negative infinity. They are determined by finding the limits of the function as x tends to ±infinity. This concept is central to understanding the end behavior of rational functions and others where the degrees of the numerator and denominator play a crucial role.

Vertical Asymptotes

Vertical asymptotes occur at specific x-values where the function becomes unbounded, typically when the denominator approaches zero while the numerator remains finite or does not cancel the zero factor. To analyze vertical asymptotes, one typically sets the denominator equal to zero and evaluates the one-sided limits as x approaches these critical points, which helps in describing the function's local behavior near these discontinuities.

Limits

The concept of limits is fundamental in calculus and provides a rigorous way to describe the behavior of functions as the input approaches a particular value or infinity. Evaluating limits, including one-sided limits near points of discontinuity or infinity, is essential to determine the asymptotic behavior and continuity properties of functions.

Piecewise Behavior due to Absolute Value

The absolute value in a function introduces a piecewise structure, as the expression inside the absolute value may change sign over different intervals. This necessitates separate analysis in each region where the function's expression inside the absolute value is non-negative or negative to accurately determine the overall behavior of the function, including its asymptotic characteristics.

Transcript

00:01 All right, so in this video, we're going to be talking about kind of a peculiar function.

00:04 It's the function given by the equation, f -fx equals absolute value of 1 minus x squared all over x times x plus 1.

00:13 And we're going to be studying the asymptotic behavior of this.

00:15 So any vertical asymptotes and any horizontal asymptotes.

00:19 And i typically like to start with the horizontal asymptotes.

00:21 So let's do that.

00:22 So to find the horizontal asymptotes, we effectively need to find the limit as x goes to infinity of f -fx, and the limit as x goes to minus infinity of f of x.

00:35 Okay.

00:36 So before i even start doing any of these limits, limit things, it might be helpful to do some sort of mathematical rearranging.

00:43 It's always a good bet when you're trying to do infinite limits.

00:46 If you don't have any nice tricks up your sleeve, it's always nice to do some algebra, right? because as it stands right now, it's not obvious what this limit is as x goes to infinity.

00:54 So let's try and manipulate a little bit.

00:56 So let's start off with our function.

00:57 I'm going to rewrite it.

00:58 But i'm actually going to rewrite it in sort of a strange way.

01:01 So it's a fact that if i have the absolute value of 1 minus x squared, well, that's the same thing as writing the absolute value of x squared minus 1, right? because the only difference between 1 minus x squared and x squared minus 1 is that they're negatives of each other, right? but if the absolute value is there, it doesn't matter if they're negatives of each other, they're going to be the same thing.

01:22 Okay? so i'm actually going to write this as x squared minus 1.

01:26 And this is for nothing other than aesthetic reasons, just because i think that it looks better.

01:30 You don't actually have to do that part of it.

01:32 Right.

01:33 And then in the usual fashion, what i'm going to try and do is i'm going to try and pull out the highest power of x that i can.

01:39 So in this particular case, it looks like my highest power of x that i'd be trying to pull out is an x squared.

01:44 So before i do that, let's actually write the bottom as a quadratic in an unfactored form.

01:50 So it's factored right now.

01:51 So let's expand those brackets.

01:53 It's going to give us x squared plus x.

01:55 And we're going to just sort of like refactor it in a different way.

01:58 So let's pull out an x squared from the bottom.

02:00 Well, that's going to give us the top, we'll stay the same for now.

02:02 On the bottom, we're going to get an x squared, one plus one over x.

02:07 Okay, nice.

02:08 So we've pulled out an x squared on the bottom.

02:11 Let's try and pull out an x squared on the top.

02:13 Now, if you're not familiar with absolute value, with doing stuff with absolute value, don't be scared, right? a lot of the absolute value stuff makes a lot of common sense.

02:22 You don't have to think about it as like a, i don't know, some particular weird function where you can't do things with it.

02:27 You can still do a lot of stuff for the absolute value.

02:29 All right, so let's try and factor out an x squared on the inside of the absolute value.

02:32 Value first.

02:34 Okay.

02:34 So pulling out an x squared from those two inner terms would give us a one leftover and a minus one over x squared.

02:40 Okay.

02:41 So we've just taken out an x squared from the top.

02:44 We've taken an x squared from the bottom.

02:45 But now we've got to deal with that pesky absolute value sign.

02:49 Well, here's the nice thing.

02:50 X squared is positive no matter what you plug into it.

02:53 Okay.

02:53 So if you're dealing with real numbers, you plug in anything into x squared, it's always positive.

02:58 And not only that, but the absolute value of a product is the product of their absolute values.

03:03 So we could write this as absolute value of x squared times the absolute value of 1 minus 1 over x squared just like this, all over x squared times 1 plus 1 over x.

03:13 And then, as i said before, since x squared is always positive, we can actually just drop those absolute value signs, right? so that first term is just going to be regular old x squared times the absolute value of 1 minus 1 over x squared.

03:25 And on the bottom, we're still going to get x squared times 1 plus 1 over x.

03:30 And so now the nice thing is we've written this in a way where we get cancellation.

03:33 So we cancel our x squared and we have an alternate way of representing f of x, f of x is going to be given by absolute value of one minus one over x squared over one plus one over x.

03:46 And so now we're actually ready to apply this to our limits.

03:48 So let's take the positive limit first.

03:50 Let's take the limit as x goes to infinity as x goes to infinity of absolute value of one minus one over x squared all over one plus one over x.

04:01 Well as is usually the case, we have some terms that die out here, right? so this 1 over x squared term and this 1 over x term, these are both going to die out and go to 0 and the limit as x goes to infinity.

04:12 Okay.

04:13 And what are we left off with? well, we're left with just absolute value of 1 over, absolute value.

04:18 I make that a little more defined over 1, which is, of course, just 1, right? no, no trickiness here.

04:24 And the very similar thing, a very similar thing is going to happen.

04:26 We do the limit as x goes to minus infinity.

04:30 So you take the limit as x goes to minus infinity.

04:32 We can write our function not exactly the way we did before, 1 minus 1 over x squared, all over 1 plus 1 over x.

04:38 And again, these terms are going to zero.

04:41 These terms are going to die out.

04:42 They're both going to go to zero.

04:43 And we're again left off with, not that interestingly, absolute value of 1 over 1, which is 1.

04:51 And so this tells us that we have horizontal asymptotes going to 1 in both directions.

04:56 Okay, so to the right and left, so we have a horizontal asymptote.

05:00 We have a horizontal asymptote we have a horizontal asymptote to the right and left i'll say in both directions given by the equation y equals 1 okay so our function is approaching 1 as we go really far off to the left and really far off to the right all right so now let's deal with the vertical asymptotes this one's a little bit tricky it's actually fairly tricky so when we're dealing with vertical asymptotes i'll just write down the function again when we're dealing with vertical asymptotes, it's really good to try and figure out if the denominator goes to zero.

05:42 And if the denominator goes to zero, does that actually necessarily mean it's a vertical ascentot or not? and the way that we do that is by seeing if we get any cancellation with the numerator.

05:50 So i'm again going to do the first step that i did in the beginning, which is i'm going to write the absolute value of 1 minus x squared as x squared minus 1 over x times x plus 1.

06:00 And now what i'm going to do is i'm going to factor the numerator, but i'm only going to factor it inside the absolute value.

06:05 Because i can't, there's not a very intuitive way of factoring absolute value.

06:10 So let's just factor it inside of the absolute value.

06:13 So i'm going to get absolute value of x plus 1 times x minus 1, right? that's because the difference of two squares and the difference of two squares are really easy to factor.

06:24 Okay, so it sort of looks like i have cancellation here with my x plus ones, but it's not exactly cancelable with that absolute value sign there.

06:32 So we've got to be a little bit more careful than that.

06:34 But it is fairly obvious here that we're going to have a vertical asymptote and x is equal to zero.

06:40 Okay.

06:41 That is fairly obvious because there's no way to get cancellation.

06:44 Even if the absolute value bars were dropped, that x on the bottom there is not going to cancel out.

06:48 So let's deal with that asymptote before we deal with the x plus one asymptote.

06:53 Okay.

06:53 So we definitely have a vertical asymptote.

06:57 We definitely have a vertical asymptote given by the equation x is equal to zero...

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