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

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the independent variable approaches positive or negative infinity. They indicate the constant value that the function approaches for very large or very small inputs and are typically determined by comparing the degrees of the numerator and denominator in a rational function.

Vertical Asymptotes

Vertical asymptotes occur at values of the independent variable where the function becomes undefined due to division by zero. They are identified by finding values that make the denominator zero (after considering any cancellations) and analyzing the one-sided limits at these points to determine if the function diverges to infinity or negative infinity.

One-Sided Limits

One-sided limits are used to analyze the behavior of a function as the variable approaches a specific point from either the left or the right. They are particularly important for identifying the behavior near vertical asymptotes, where the function can approach different infinite values depending on the direction of approach.

Limit Evaluation Techniques

Limit evaluation techniques are essential for determining asymptotic behavior and include methods such as algebraic manipulation, canceling common factors, and comparing the highest power terms in the numerator and denominator of rational functions. These techniques help in finding both horizontal asymptotes through limits at infinity and vertical asymptotes via one-sided limits.

Transcript

00:01 Alright, so in this video, we're going to talk about the function f of x equals x squared minus 9 over x times x minus 3.

00:07 And we're particularly interested with identifying any asymptotes that it has, right? so, horizontal and vertical asymptotes.

00:13 That's our goal.

00:14 So let's start with the horizontal asymptotes.

00:16 How do we find horizontal asymptotes? well, the technique is always pretty similar, right? we want to find the limit as x goes to infinity of f of x, and we want to find the limit as x goes to minus infinity.

00:31 Of f of x right this is going to tell us about the end behavior of the function what happens you know in the tail to the right and the tail to the left of this function what is it what does it approach if it approaches anything all right but before we actually dive into to finding any either these limits it might be helpful to do some algebraic simplification right if you can always if you can simplify your function it's always best too right if you can see how to simplify quickly so in this particular case let's write our function down again what we notice is that the numerator can be factored right it's a different of two squares.

01:03 It's a difference of the, it's a difference of x minus, it's a product of x minus 3 and x plus 3, right? so that's, it's helpful to know your difference of two squares, those those come up quite a bit.

01:14 And if not just knowing how to factor quadratics, right? so the numerator can be written as x minus 3 times x plus 3.

01:20 The denominator will write the exact same way.

01:22 And writing it in this form, we see that we actually get some cancellations, right? so x minus 3 cancels with x minus 3.

01:28 And so we can actually write this function in a slightly simpler form, we read it as x plus 3 over x.

01:35 And so this is nice because it's quite a bit easier to look at than the original function.

01:40 It's especially going to be useful when identifying our vertical asymptotes.

01:43 But it's actually really helpful for finding our horizontal ones as well.

01:46 Okay, so if we're now trying to find the limit of f of x as x goes to infinity, it's good enough to just find the limit as x goes to infinity of x plus 3 over x.

02:01 Okay.

02:01 Now, some of the ones of you who are more practice will know that this limit's going to equal one immediately but if you actually want to go through the steps in algebra what you would do here is you factor out the highest power of x from both the numerator and the denominator in this case the highest power of x in both of them is just x right so we would alternatively write this as this is sort of silly in this particular case, but it's okay to write it we'd factor out an x and what we have left over is a one and a three over x and then the bottom we'd factor out an x and what we'd have left over is just well, a one.

02:33 Nothing too interesting there.

02:36 All right, and so we get another cancellation.

02:39 The x has canceled each other out.

02:40 And what we're left with now is trying to find the limit as x goes to infinity of just 1 plus 3 over x, all over x.

02:49 Sorry, all over 1.

02:51 Not over x.

02:52 All over 1.

02:54 And so this is a much easier limit to take because we can see that this term right here, this term is dying out to zero as x gets really, really big.

03:02 And so we're left with just the limit of 1.

03:05 All right.

03:05 So as we go off really, really far to the right, this function gets closer and closer to 1.

03:10 Let's see what happens when we go off to the left.

03:12 And so if we take the limit as x goes to minus infinity of fx, well, all the same steps in algebra are good up till here, right? so we can actually still say that finding the limit of f of x as x goes to minus infinity is the same thing as finding the limit as x goes to minus infinity of 1 plus 3 over x over 1.

03:32 Right the steps that we followed in the above line in algebra those are all going to be the exact same right and again this term right this 3 over x term that's going to go to zero right it's going to zero from the left but it's still going to zero right so this trails off to zero as x goes to minus infinity and we're again left with just a limit of one and so what this is telling us is that our function is going to approach one as we go to the right and it's going to approach one as we go to the left and so we have a horizontal we have one horizontal asymptotes so we have one horizontal asymptote, and it's y equals one, right? the equation is y equals one.

04:13 All right, so let's discuss the vertical asymptotes a little bit.

04:16 Now, we did some algebraic simplification in the very beginning, right? we had written our function as x plus three over x.

04:22 Let's use that representation.

04:24 Now, technically, if we were talking about something called singularity here, there are singularities, meaning problem points.

04:31 Technically, x equals three is a singularity as well, because we have an x minus three factor in our original representation, but it's something that's called a removable singularity.

04:41 And it's removable because it cancels out with a similar factor in the numerator.

04:45 So for this particular problem, we aren't going to consider that one at all.

04:48 We're just going to consider our simplified version of our function...

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