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Please-i-need-someones-help-on-this-How-do-i-find-an-Asymptote-to-a-curve-and-also-how-find-a-general-solution-for-a-differential-equation-

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Question Number 63983 by Rio Michael last updated on 11/Jul/19
Please i need someones help on this How do i find an Asymptote to a curve? and also how find a general solution for a differential equation.
$${Please}\:{i}\:{need}\:{someones}\:{help}\:{on}\:{this}\: \\ $$$${How}\:{do}\:{i}\:{find}\:{an}\:{Asymptote}\:{to}\:{a}\:{curve}? \\ $$$${and}\:{also}\:{how}\:{find}\:{a}\:{general}\:{solution}\:{for}\:{a}\:{differential}\: \\ $$$${equation}. \\ $$$$ \\ $$
Answered by MJS last updated on 12/Jul/19
asymptotes f(x): y=(1/x) is not defined for x=0 ⇒ the line x=0 is an asymptote now let′s look at the inverse (solve y=(1/x) for x) f^� (y): x=(1/y) similar as above, this is not defined for y=0 ⇒ ⇒ the line y=0 is also an asymptote f(x): y=((x−1)/(x+1)) not defined for x+1=0 ⇒ asymptote x=−1 f^� (y): x=((y+1)/(1−y)) not defined for 1−y=0 ⇒ asymptote y=1 f(x): y=((x+1)/((x−1)(x+2))) not defined for x+2=0 and x−1=0 ⇒ 2 asymptoted x=−2 and x=1 f^� (y): x=−((y−1±(√(9y^2 +2y+1)))/(2y)) not defined for 2y=0 ⇒ asymptote y=0 f(x): y=(((x−1)(x+2))/(x+1)) not defined for x+1=0 ⇒ asymptote x=−1 f^� (y): x=((y−1±(√(y^2 +2y+9)))/2) defined ∀y∈R ⇒ no asymptote but f(x): y=(((x−1)(x+2))/(x+1))=x−(2/(x+1)) ⇒ we have another asymptote y=x same here: f(x): y=((3x^2 +x+2)/(2x−5l1))=(3/2)x+(5/4)+((13)/(4(2x−1))) not defined for 2x−1=0 ⇒ asymptote x=(1/2) plus asymptote y=(3/2)x+(5/4) f^� (y): x=((2y−1±(√(4y^2 −16y−23)))/6) this is defined ∀y∈R, although x∉R for some values of y. (√(4y^2 −16y−23)) might is not always a real number, but it′s always ∈C. this is different from terms like ((term)/(x+a)) which are not defined for x+a=0 in any set if numbers we can also have asymptotic curves f(x): y=(((x+1)(x+2)(x+3))/x)=x^2 +6x+11+(6/x) has an asymptote x=0 plus the asymptotic curve y=x^2 +6x+11
$$\mathrm{asymptotes} \\ $$$$ \\ $$$${f}\left({x}\right):\:{y}=\frac{\mathrm{1}}{{x}} \\ $$$$\mathrm{is}\:\mathrm{not}\:\mathrm{defined}\:\mathrm{for}\:{x}=\mathrm{0}\:\Rightarrow\:\mathrm{the}\:\mathrm{line}\:{x}=\mathrm{0}\:\mathrm{is}\:\mathrm{an} \\ $$$$\mathrm{asymptote} \\ $$$$\mathrm{now}\:\mathrm{let}'\mathrm{s}\:\mathrm{look}\:\mathrm{at}\:\mathrm{the}\:\mathrm{inverse}\:\left(\mathrm{solve}\:{y}=\frac{\mathrm{1}}{{x}}\:\mathrm{for}\:{x}\right) \\ $$$$\bar {{f}}\left({y}\right):\:{x}=\frac{\mathrm{1}}{{y}} \\ $$$$\mathrm{similar}\:\mathrm{as}\:\mathrm{above},\:\mathrm{this}\:\mathrm{is}\:\mathrm{not}\:\mathrm{defined}\:\mathrm{for}\:{y}=\mathrm{0}\:\Rightarrow \\ $$$$\Rightarrow\:\mathrm{the}\:\mathrm{line}\:{y}=\mathrm{0}\:\mathrm{is}\:\mathrm{also}\:\mathrm{an}\:\mathrm{asymptote} \\ $$$$ \\ $$$${f}\left({x}\right):\:{y}=\frac{{x}−\mathrm{1}}{{x}+\mathrm{1}} \\ $$$$\mathrm{not}\:\mathrm{defined}\:\mathrm{for}\:{x}+\mathrm{1}=\mathrm{0}\:\Rightarrow\:\mathrm{asymptote}\:{x}=−\mathrm{1} \\ $$$$\bar {{f}}\left({y}\right):\:{x}=\frac{{y}+\mathrm{1}}{\mathrm{1}−{y}} \\ $$$$\mathrm{not}\:\mathrm{defined}\:\mathrm{for}\:\mathrm{1}−{y}=\mathrm{0}\:\Rightarrow\:\mathrm{asymptote}\:{y}=\mathrm{1} \\ $$$$ \\ $$$${f}\left({x}\right):\:{y}=\frac{{x}+\mathrm{1}}{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right)} \\ $$$$\mathrm{not}\:\mathrm{defined}\:\mathrm{for}\:{x}+\mathrm{2}=\mathrm{0}\:\mathrm{and}\:{x}−\mathrm{1}=\mathrm{0}\:\Rightarrow\:\mathrm{2}\:\mathrm{asymptoted} \\ $$$${x}=−\mathrm{2}\:\mathrm{and}\:{x}=\mathrm{1} \\ $$$$\bar {{f}}\left({y}\right):\:{x}=−\frac{{y}−\mathrm{1}\pm\sqrt{\mathrm{9}{y}^{\mathrm{2}} +\mathrm{2}{y}+\mathrm{1}}}{\mathrm{2}{y}} \\ $$$$\mathrm{not}\:\mathrm{defined}\:\mathrm{for}\:\mathrm{2}{y}=\mathrm{0}\:\Rightarrow\:\mathrm{asymptote}\:{y}=\mathrm{0} \\ $$$$ \\ $$$${f}\left({x}\right):\:{y}=\frac{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right)}{{x}+\mathrm{1}} \\ $$$$\mathrm{not}\:\mathrm{defined}\:\mathrm{for}\:{x}+\mathrm{1}=\mathrm{0}\:\Rightarrow\:\mathrm{asymptote}\:{x}=−\mathrm{1} \\ $$$$\bar {{f}}\left({y}\right):\:{x}=\frac{{y}−\mathrm{1}\pm\sqrt{{y}^{\mathrm{2}} +\mathrm{2}{y}+\mathrm{9}}}{\mathrm{2}} \\ $$$$\mathrm{defined}\:\forall{y}\in\mathbb{R}\:\Rightarrow\:\mathrm{no}\:\mathrm{asymptote} \\ $$$$\mathrm{but} \\ $$$${f}\left({x}\right):\:{y}=\frac{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right)}{{x}+\mathrm{1}}={x}−\frac{\mathrm{2}}{{x}+\mathrm{1}}\:\Rightarrow\:\mathrm{we}\:\mathrm{have} \\ $$$$\mathrm{another}\:\mathrm{asymptote}\:{y}={x} \\ $$$$ \\ $$$$\mathrm{same}\:\mathrm{here}: \\ $$$${f}\left({x}\right):\:{y}=\frac{\mathrm{3}{x}^{\mathrm{2}} +{x}+\mathrm{2}}{\mathrm{2}{x}−\mathrm{5l1}}=\frac{\mathrm{3}}{\mathrm{2}}{x}+\frac{\mathrm{5}}{\mathrm{4}}+\frac{\mathrm{13}}{\mathrm{4}\left(\mathrm{2}{x}−\mathrm{1}\right)} \\ $$$$\mathrm{not}\:\mathrm{defined}\:\mathrm{for}\:\mathrm{2}{x}−\mathrm{1}=\mathrm{0}\:\Rightarrow\:\mathrm{asymptote}\:{x}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{plus}\:\mathrm{asymptote}\:{y}=\frac{\mathrm{3}}{\mathrm{2}}{x}+\frac{\mathrm{5}}{\mathrm{4}} \\ $$$$\bar {{f}}\left({y}\right):\:{x}=\frac{\mathrm{2}{y}−\mathrm{1}\pm\sqrt{\mathrm{4}{y}^{\mathrm{2}} −\mathrm{16}{y}−\mathrm{23}}}{\mathrm{6}} \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{defined}\:\forall{y}\in\mathbb{R},\:\mathrm{although}\:{x}\notin\mathbb{R}\:\mathrm{for}\:\mathrm{some} \\ $$$$\mathrm{values}\:\mathrm{of}\:{y}.\:\sqrt{\mathrm{4}{y}^{\mathrm{2}} −\mathrm{16}{y}−\mathrm{23}}\:\mathrm{might}\:\mathrm{is}\:\mathrm{not}\:\mathrm{always} \\ $$$$\mathrm{a}\:\mathrm{real}\:\mathrm{number},\:\mathrm{but}\:\mathrm{it}'\mathrm{s}\:\mathrm{always}\:\in\mathbb{C}.\:\mathrm{this}\:\mathrm{is} \\ $$$$\mathrm{different}\:\mathrm{from}\:\mathrm{terms}\:\mathrm{like}\:\frac{\mathrm{term}}{{x}+{a}}\:\mathrm{which}\:\mathrm{are}\:\mathrm{not} \\ $$$$\mathrm{defined}\:\mathrm{for}\:{x}+{a}=\mathrm{0}\:\mathrm{in}\:\mathrm{any}\:\mathrm{set}\:\mathrm{if}\:\mathrm{numbers} \\ $$$$ \\ $$$$\mathrm{we}\:\mathrm{can}\:\mathrm{also}\:\mathrm{have}\:\mathrm{asymptotic}\:\mathrm{curves} \\ $$$${f}\left({x}\right):\:{y}=\frac{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)}{{x}}={x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{11}+\frac{\mathrm{6}}{{x}} \\ $$$$\mathrm{has}\:\mathrm{an}\:\mathrm{asymptote}\:{x}=\mathrm{0}\:\mathrm{plus}\:\mathrm{the}\:\mathrm{asymptotic} \\ $$$$\mathrm{curve}\:{y}={x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{11} \\ $$
Commented by Rio Michael last updated on 12/Jul/19
God bless you sir.
$${God}\:{bless}\:{you}\:{sir}. \\ $$

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