Can we say something about roots of quadatric equation whose coefficient consists of terms that tend to 0?I am particularly interested in discussing the solution of ϵ1x2+(ϵ2+a)x+(ϵ3+b)=0 where a&gt;0,b&gt;0 are constants and ϵ1,ϵ2,ϵ3→0. Looking at the equation , I can think that one solution is −ba and other solution must tend to ∞ (similar things happen for ϵx2+ax+b=0). But, how can i prove this mathematically?

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Can we say something about roots of quadatric equation whose coefficient consists of terms that tend to 0?I am particularly interested in discussing the solution of ϵ1x2+(ϵ2+a)x+(ϵ3+b)=0 where a&amp;gt;0,b&amp;gt;0 are constants and ϵ1,ϵ2,ϵ3→0. Looking at the equation , I can think that one solution is −ba and other solution must tend to ∞ (similar things happen for ϵx2+ax+b=0). But, how can i prove this mathematically?

stadfeste8ru · Accepted Answer

Clearly, for any&amp;nbsp;ϵ1≠0, the solutions of your quadratic equation arex1,2=12ϵ1(−(ϵ2+a)±(ϵ2+a)2−4ϵ1(ϵ3+b)).For&amp;nbsp;ϵ1→0, we can usex+a=x1+ax=x(1+a2x+O{(ax)2})&amp;nbsp;for&amp;nbsp;axll1&amp;nbsp;and thusx1,2=12ϵ1(−(ϵ2+a)±(ϵ2+a)(1−2ϵ1(ϵ3+b)(ϵ2+a)2+O{(ϵ1(ϵ3+b)(ϵ2+a)2)2})).For&amp;nbsp;a≠0&amp;nbsp;the first-order expansion becomes asymptotically exact for&amp;nbsp;ϵ1→0&amp;nbsp;and we thus havex1→−12ϵ12ϵ1(ϵ3+b)ϵ2+a=−ϵ3+bϵ2+a,converging to&amp;nbsp;−ba. For&amp;nbsp;x2&amp;nbsp;we havex2→−12ϵ1(−2(ϵ2+a)+2ϵ1(ϵ3+b)ϵ2+a)=ϵ2+aϵ1−ϵ3+bϵ2+a,which diverges for&amp;nbsp;ϵ1→0&amp;nbsp;provided that&amp;nbsp;a≠0.

Can we say something about roots of quadatric

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