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

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02:41

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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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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