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Find the horizontal and vertical asymptotes of ea…

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Problem 49 Easy Difficulty

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{2x^2 + x - 1}{x^2 + x -2} $

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Video Transcript

for this problem we want to find the horizontal and vertical sm totes of the curve. Why equal to two X squared plus x minus one over X squared plus x minus two for the horizontal s in total to find it. We would take the limit of the function or the curve as X approaches positive and negative infinity. And so from here we have Why equal to the limit? As X approaches plus minus infinity of two, x squared plus X -1 over Extra Plus X -2. Now, since we're dealing with limits of infinity then we would factor out the variable, the highest exponent for the numerator and variable with the highest exponents for the denominator. And so and here we have the limit as X approaches plus, remind us infinity of we have x squared times two plus one over x minus one over X squared this all over X squared times we have one plus one over x minus two over x squared simplifying We have limit as X approaches plus or minus infinity of two plus one over x minus one over x squared this divided by one plus one over x -2 over x squared. Now evaluating at infinity we have two Plus 1 over plus minus infinity minus. We have plus or minus infinity over one plus 1 over plus, remind us infinity minus two over plus or minus infinity. Now constant over infinity will always approach zero And so this term of is zero. So as this one also this one and this one and so we are left with 2/1 or two. Therefore our horizontal sm toad Is y equal to two. How to find the vertical ascent. Oh, it was simply Set the denominator equal to zero and solve for X. That means we solve for expert plus X minus two equals zero. Now we can factor this and we have X Plus two times X -1. This is equal to zero and this will give us Exports to equal zero Or X -1 equals zero. This is just x equals negative two and X equals one. Therefore the vertical essen toads are X equals -2 and X equals one. Over here we have the graph of the function and from here we can see that the horizontal sm toad is indeed Y equals two. And the vertical lesson toads are X equals negative two And this one is x equals one.

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Calculus: Early Transcendentals

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