Problem

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

Problem 50 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
$$f(x)=m x+b+R(x) / Q(x) ]$$
$$y=\frac{1+5 x-2 x^{2}}{x-2}$$

Answer

$(-\infty, 2)$

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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 way are given a function. F of X is equal to the ratio or the fraction off eggs through the third minus two. Great plus one on this is X squared and they want us to find a slang as in third a slam aceptar Ah, once again, what's the slant? Aceptar It is aligned. Why equals m x plus v such that the limit when x goes to infinity off the difference between f of X and that line the election is zero or or and the limit when it goes to minus infinity off this lot is equal to zero. So the hint given is that we should use division divide the numerator by the denominator on. From there you will see which limiter function will be would be appropriate to take as slant aesthetic. Okay, so there is only one term in the denominator, so we can say that f of X is equal to x third Liberec squared minus two X squared career plus one over X squared. Some canceling is going to happen here. Canceling is going to happen here and we will have the f of X is equal to my two plus one over X squared now because this will go to zero when it tends to plus or minus infinity then for R m explosively, we take this to be both me and we say that our slant essence there is going to be Why equals X miles? Okay, this is the answer to the question. What is the slight s and third off this function on What we're going to do now is check this in a in a You know, graphic calculators such as this was calm. We'll say that if they is the ratio off to the third minus who to the second plus form over? Okay, to think. Okay, on we say that why Equals explains to waas are wolf. Our let's see was exposed to Okay. What is happening is from the graph we can tell that from this graph we can see and confirmed that this really is true that this is this life essence 30. Why, right? So when it girls tends towards the far right in, then the function f phobics. You see this? What is this color? This is purple. The purple graph is the graph off If and as X moves away from 23456 And so on The we yes, zoom square here and then zooming to see what happens. You see what happens As ex grows and becomes greater and greater, the purple graph becomes closer and closer to the to the graph off. Ah, this line which means that this is in fact this land The slam s entered on clean. All right, so but difference between F Away and X minus two becomes minuscule here, which means that as X grows further to the right, it will be zero. And furthermore, when it goes to minus infinity, we go that way. F of X also nears the nears the breath off the line. Why call six months? So this here is the slam essence off this function on. Then we have the whole solution of the problem. Hope it helps. Bye.

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