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Answer on Calculus Question for adriana

Question #4797
consider the function F=
x^2 | x+2| for x<-2
(2/x)+1 for -2 <or= x < 0
3 for x=o
(sinx/x) for x>0

describe all asymptotes of F (vertical and horizontal)
Expert's answer
Asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity.We have function
data:image/png;base64,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
data:image/png;base64,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
The line x=a is a vertical asymptote of the function if at least one of the following statements is true:
data:image/png;base64,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
Function is defined everywhere except x=0
Let’s find limit
data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARoAAAAxCAIAAACKzfGsAAAE50lEQVR4nO2c23nkIAyFp4F93hZSg7tIE/SRDmhg2/BTOnEx7IMvXCVkIxh7vvM/JZmxQUZHCFD8cgAAJV7v7gAAnwPkBIAakBMAakBOYCCzeb3M/Pw2KCAnMIiRbj6b12SXMW2FQE5gBIudSlKazWtDXWiLncYrCnIC/SmLyf91sVMPRc1mtKAgJ9Abamayga8vdvqEGQpyAp1ZrBG4dKeV1eAJCnICfZGpabFTn32KsXqCnEBXiEwv/1Yvrx+qJ8gJdGU2AjX1dHmZnpWAnEBPBKleb38fOT1BTqAnP19/DevLobMv1nbQ1e/33+mf/m2LQE6gJz9fXz/0p/4Ut9NZrnPO/X7/4fqgCimn9WStQDRzzoZ7BPynEub5TcVXK8dDeFcN2OOh5VRwsD5P+RZycm4NHpF6og0YyUnBYqfrxVPvrGWM25f05PiOQrdlscz1D2eu1S5+dlJjsVP+cDbuIqeCFvw+jXiorp8o9JVTbYUa97tSU7llLVpy2m9JxzJhO03hzCnYNUJOe8p4czmFLpSsEk9V7AoEFSfRZo7+0klSvKsln1b9UrtKho1lbkg4Oy5vuF5TTtwWIDM+N5FTIJlCYMz6vvn//vf11/W3yojGIzYbaThkWpTBpQhp47XOaMuJiWVOP5zxFzfYJTp2knflipxaNuL3rHuyS5SCUzek5ZTsushcax25bPxO5QnnknWiRSHbEypcel1O+ze3bG0fBDP7Aal3lYllrqAmPrK06KnRrkfLyV85m8kYH3vJIg5STqH7JFdzAasc8LN1QPm6RLwyETJTTLoPy5E2lDReDdLbF/bFhj3smewSu3v9YbCxjH4wdGQpXEA+m+R7jXbdUk5C26MjaJ805Z95KDnFAxDV0vOuVYyEvAetUjo+P7uV1Bx7Sd+8tHbKf3D5zxU5MbHMMc+fjCwSBfexq1lOYtfXn52iHDtNCGJJ7BBy4p8/l+wZY0TRMbwmc9xTyV6xRRHc2ilWaV0C59yO7y8Xy9gbUJFlmJwKI1/sKJ8p0F0ZmuyFM1AyOZ2Tk2AToBwEJ7sc4dT3pjJ/hIMdTheiHcFyixKqt/dWCoStKKeq85PJHhVZ2pM9ZTldYnSy569cxWT8Zip1ypLJKVzEsPNTtpqKn+uplXC0K+6HJb1R3k2iRQmSysjjYQiXcCpyEmi3vLNKRhaVrYh7y4kJQS07e8mZ1uGn1A0v1+zJt2pVzuZvDrfpESzfs6V8giyWuXQbvRJZGtTUbNeIc6ewk4WndpNzpxqCJfX7C4U+loeEs0FFRhwPkZNzQ2rGAMEjwhnkBMaQnA9c4nbhLDVKc+10kaa100kgpzexD/L4d8F1JDcKcgKX2VKrYoYVj2pUv3JvPXFGuciuglE3sO7c8UkbkJMys53IYBgkX+Gh3LjRvgpnlPN2FYzirWsuYpaAd0U8mMN/+LPCB81OrlAfULarZFQ926NLDZX6jjcZPRY2jodbA09aO1UmJ29XwSiBN7O1Xs3gPXsPZQuwxAF95ld9UhtteKNcaldulOzVYN22IPEWWPBR1PTUUsRcBe8oBx8Gm++1FDHXGb1PDzmB/pTOk9uLmAXNjk6lIScwgqH7ayvv2OSBnMAghpY8valcFHIC48j+p/WxjRBATgCoATkBoAbkBIAakBMAakBOAKgBOQGgxn9xi0XT/yM6TgAAAABJRU5ErkJggg==
Therefore we have the vertical asymptote x=0
Horizontal asymptotes are horizontal lines that the graph of the function approaches as
data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEAAAAAcCAIAAABeRy4FAAAA8ElEQVRYhe2W0Q2EMAiGmasD/fMwDcswjPdQa21tlNZLuEv4H0xQoXyARNr+XOSdwFsFgLcCwFsB4K0AmJZygnwvXAAUCRKr5cVfBdiUE1kYBgDKiYgouxeDiCygJwBBjXKYxrLWPMYOe+yrICciQQJqDOX0fHzfAeVEkHw1Z94iPFSu74AyapaCxr15NtZlhFbKOK6sGYCrKWjPFp4G2FarX3zfdKBrwAqAMoBFgpvmtW+10Y8bOX3gON2y2VqA7FE+HsMEdumvdq4MZ873GNWJLdT2frdmNtBDte620IwEvUv8C3krALwVAN4KAG99AJ1H052gcumQAAAAAElFTkSuQmCC
data:image/png;base64,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
Therefore we have the horizontal asymptote y=1
When a linear asymptote is not parallel to the x- or y-axis, it is called an oblique asymptote or slant asymptote.
A function f(x) is asymptotic to the straight line
data:image/png;base64,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
if
data:image/png;base64,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
Under the function schedule we see that others asymptotes are not present.
In other points it isn’t asymptotes because there function is continuous.

We have two asymtotes x=0(vertical) and y=1(horizontal).

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