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Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.

$ y = \dfrac{x^3 - x}{x^2 - 6x + 5} $

$x=5$

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This is problem number fifty one of the Stuart Calculus eighth Edition Section two point six. Find the horizontal and vertical ascent ots of each curve. If you have a graphing device, chicken work by graphing the curve and estimating the Assam toots and the curve of the function is why equals? Execute minus X. Divided by the quantity X squared minus six X plus five For our vertical asthma, too, we asked ourselves, What is the denominator? What values make the denominator? Ah, equal to zero. So we go ahead and try it out. X squared minus six X plus five equals zero. One is this. We go out and factor quadratic. We can infect their into two point no meals. I mean, it's fine and this one and this kisses X equals and X equals one. Before we can confirm whether these air vertical attitudes, we also have to simplify the numerator. In case either of these excise are removable at X equals five and X equals one. We know there is a discontinuity because they make that a nominee who's nominator zero. However, it's ah possible that it's a removal discount If the U numerator shares affect er, so if we simplified the writer or get extends X squared, Mintz won and denominator, we still have X minus five times X minus one and simplify a little further. This is a difference of squared X squared minus one and express one x plus one. And the denominator means sing and we see that X equal to one is actually the removable dis continuity. And since the experience one term cancels Therefore, for the sake of this problem, there's only one vertical as him too Vertical asked me to, and that is that the only word closet is that X equals five. And this is the equation of that. Hasn't o X equals five? Thanks, Eagle one is removable. It'LL just be a hole in the function. It will not be a vertical as some tote for horizontal aspic dudes, we take the function and we ask ourselves what is the limit? As experts infinity and the function is X cubed minutes Thanks two hundred bunny X squared by six x plus fire. One step we can take is to her the body's turned by X squared. Go ahead and do that. Execute over X squared is X minus X R X Squared is one of ex didn't know me with X squared or X squared which is one six x or X squared which is six Rex and then vital work squared is the last term as a re approach Infinity Each of these terms wanna rex six ranks and February X squared Go to zero and we see that the limit beverages towards infinity Since the remaining parts are ex over one and X going towards infinity means system it does not exist it ever just towards affinity. And this confirms that we indeed do not have a horizontal has some tote For this function we can also check negative infinity The limit is this experts negative infinity and we get the same results except the limit Beverages too negative infinity therefore way include thie function has no horizontal asked. But earlier we did find one Britta classes at X equals five and we can use a graphing tool and plot the function. And here we can see clearly there's a vertical ASM two at X equals five, but there is no horizontal asked me to on this is in agreement with our work for this problem