Problem

$49-52=$ The line $y=m x+b$ is called a slant asy…

Problem 49 Easy Difficulty

$49-52=$ The line $y=m x+b$ is called a slant asymptote
if $f(x)-(m x+b) \rightarrow 0$ as $x \rightarrow \infty$ or $x \rightarrow-\infty$ because the
vertical distance between the curve $y=f(x)$ and the line
$y=m x+b$ approaches 0 as $x$ becomes large. Find an equation of the slant asymptote of the function and use it to help
sketch the graph. [ For rational functions, a slant asymptote
occurs when the degree of the numerator is one more than the
degree of the denominator. To find it, use long division to write
$$f(x)=m x+b+R(x) / Q(x) ]$$
$$y=\frac{x^{2}}{x-1}$$

Answer

$(-\infty, 1)$

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Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 4

Curve Sketching

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Video Transcript

Hello. Look, given the function eggs the power of 31 swarm on X squared My swim. We want to find a slave. A symptom. Ask him to three instructions given by the text are a scholar So slight Essence. That is a line that has, uh, in equation. Why was MX plus be such that the limit went extends to plus infinity off the difference between the function values and the values off the slam s in the palace zero or and the same limits. When X goes to minus infinity off the same thing is equal zero. What this basically means is that if we have a line and this happens, then X goes to infinity. And if off X goes towards the Y off the line that we have here, the hint is the hint is to use long division that is a polynomial on. That is what we're going to do. We're going to divide extracts of the third with minus one. I'll explain why I've written here just in a second X squared, minus one. When we do polynomial division, then we want Thio truth. The the there are missing missing missing terms off the polynomial. So we list all the terms here we have 00 We could have just lifted a zero just as an empty space. So what do our multiply X square to get X to the power of three with x x times X squared is X to the power of three minus x. I write it here. Okay, Andi, I have zero x squared and I have changed the I changed the signs because we will subtract, uh, these two terms from the terms about so these to annihilate each other. 00 zero plus six is just x by itself X once one. Now we have arrived. Uh huh. The remainder, that is, that has a degree list than our divisor is. So we say that if a vic is equal to the cautions, which is a plus x y the swung over X squared minus one. No, If we have a look at this, we will see that the limit went in when X goes to infinity off. This is we'll rewrite this as a difference of squares so we can cancel these two. So when X is a huge number, the numerator is one on the denominator is a huge number, which means zero. So if we say that this is X See here this is MX Busby. Then we will have f of X minus X will give us this fraction. We've just found out that the limit off distraction zero. So why equals? It is our slept is our slam s entered. What we need to do now is to check with our graphical calculator. So it's the third minus one. Thanks. Third minus one. Um over to the second, minus one on we claim that are slant essence, that is. And is it so? Okay, let's change. So, like this to be read. All right, So this is the the graph off f of X and we see as X increases as it moves towards plus affinity. The the graph off a few weeks is ever so closer to the breath or this line. Why minus which basically says that why equals X is the slight s. That's what we needed to show. Right? So when extends to infinity than every week, the red right seems to near the graph off the slant s center. Why equals a the limit off a four weeks. Months equals zero. Why? Because F of X minus six is this fraction. Here In this fraction, we have have the era, which means we have answered the questions. Solve the problem. Hope it helps, but

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