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A function $f$ has a slant asymptote $y=m x+b(m \neq 0)$ if $\lim _{x \rightarrow \infty}[f(x)-(m x+b)]=0$ and/or $\lim _{x \rightarrow-\infty}[f(x)-(m x+b)]=0$In exercises $43-48,$ find the slant asymptote. (Use long division to rewrite the function.) Then, graph the function and its asymptote on the same axes.$$f(x)=\frac{3 x^{2}-1}{x-1}$$

$y=3 x+3$

Calculus 1 / AB

Chapter 3

Applications of Differentiation

Section 6

Overview of Curve Sketching

Derivatives

Differentiation

Applications of the Derivative

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

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In mathematics, precalculu…

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In mathematics, a function…

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A function $f$ has a slant…

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Assume that $f(x)$ has dom…

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Hello. Hello. Hello. We have f of X defined to be three x grade minus juan over X minus one. We need to find the slam slammed s in third this land as symptoms which will be a line with with non zero slope MX plus Be such the limit when x goes either to plus infinity or minus infinity off if off minus that m explosively two x possibly we want the limit off this V zero. If this happens, then what we're reading from here is as X moves to the far right than the then the mm. The graph approaches approaches some line, um, expo speak or if it moves to the far left, then the graph will approach some slant line that is some lines with with slope M and interested be. The hint tells us the hint tells us is divide or use long division or rewrite this in a way that will help you find that and then your function. So three X squared minus one, divided by Let's let's divide. And here we have three x squared minus. That is plus zero. There's no excess minus one. We could have done this with synthetic division, but whatever. So what are we divide what we multiplied this with We multiply this three x three x times x three x squared three X times minus one is one is three Change the size. So this this annihilate zero on because becomes three x we lower the risk of polynomial down whether we multiply the leading factor with plus 33 times x is three x three times months one is minus for reid. So change signs. Uh, um, I really and we have a left over to Therefore, uh, a few weeks is equal to, um mm is equal to through X minus One is equal through the quote three X plus three and plus Q over the X minus. One way we're dividing this with because if we multiply, if we multiply this predicts one swarm we would have three X plus three that multiply X minus one plus two, which is the way that we check our if our division is correct. So we multiplied divisor with the So this is our M expressivity. Our slant, a flight s center for this function is going to be three x plus for real. We will check this by going to our calculator. Um, so we have f of X is three minus one over x months on their slammed essence. Okay, the red graph is the graph of the function of the green graph is the line that represents the essence that as big as picks tends towards plus infinity. Uh huh. Well, extends towards plus ability. All right, so way have when x goes towards plus infinity, then a few weeks nears. The function nears the graph of three X plus three. Okay, this is it. So when it goes to the far right of the graph off, if comes ever so closer to the graph off three explore three, which is in fact, the interpretation off this equation here and even if it says or or end the same limit. But when X goes to minus infinity So when extends towards violence, affinity when it goes to the left firing off the grass, we also have that effect, gets closer and closer to the graph off three for three, which is the definition off a slam essence. Flint. The solution is straight for three hope the Phillips, but

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