Evaluate the following limits: _ x x^2+a^2 + x^2+b^2 x^2+c^2 + x^2+d^2

_ x x^2+a^2 + x^2+b^2 x^2+c^2 + x^2+d^2 Rationalising the numerator and the denominator: = _ x [ ( x^2+a^2 - x^2+b^2 ) ( x^2+c^2 - x^2+d^2 ) ( x^2+c^2 - x^2+d^2 ) ( x^2+c^2 - x^2+d^2 ) ( x^2+a^2 - x^2+b^2 ) ( x^2+a^2 - x^2+b^2 ) ] = _ x [ ( x^2+a^2 - x^2+b^2 ) ( x^2+a^2 - x^2+b^2 ) ( x^2+c^2 - x^2+d^2 ) ( x^2+c^2 - x^2+d^2 ) ( x^2+c^2 + x^2+d^2 ) ( x^2+a^2 + x^2+b^2 ) ] = _ x (x^2+a^2 )- (x^2+b^2 ) (x^2+c^2 )- (x^2+d^2 ) ( x^2+c^2 + x^2+d^2 x^2+a^2 + x^2+b^2 ) = _ x ( a^2-b^2 c^2-d^2 ) ( x^2+c^2 + x^2+d^2 x^2+a^2 + x^2+b^2 ) Dividing the numerator and the denominator by x: = _ x ( a^2-b^2 c^2-d^2 ) 1+ a^2 x^2 + 1+ a^2 x^2 1+1 a^2 + 1+ b^2 a^2 As x ,1 x ,1 x^2 0 = ( a^2-b^2 c^2-d^2 ) ( 1+1 1+1 ) = a^2-b^2 c^2-d^2

Evaluate the following limits: _ x x^2+a^2 + x^2+b^2 x^2+c^2 + x^…

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Question

Evaluate the following limits:
$\lim\limits_{\text{x}\rightarrow\infty}\frac{\sqrt{\text{x}^2+\text{a}^2}+\sqrt{\text{x}^2+\text{b}^2}}{\sqrt{\text{x}^2+\text{c}^2}+\sqrt{\text{x}^2+\text{d}^2}}$

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Answer

$\lim\limits_{\text{x}\rightarrow\infty}\frac{\sqrt{\text{x}^2+\text{a}^2}+\sqrt{\text{x}^2+\text{b}^2}}{\sqrt{\text{x}^2+\text{c}^2}+\sqrt{\text{x}^2+\text{d}^2}}$
Rationalising the numerator and the denominator:
$=\lim\limits_{\text{x}\rightarrow\infty}\Bigg[\frac{\big(\sqrt{\text{x}^2+\text{a}^2}-\sqrt{\text{x}^2+\text{b}^2}\big)}{\big(\sqrt{\text{x}^2+\text{c}^2}-\sqrt{\text{x}^2+\text{d}^2}\big)}\times\frac{\big(\sqrt{\text{x}^2+\text{c}^2}-\sqrt{\text{x}^2+\text{d}^2}\big)}{\big(\sqrt{\text{x}^2+\text{c}^2}-\sqrt{\text{x}^2+\text{d}^2}\big)}\times\frac{\big(\sqrt{\text{x}^2+\text{a}^2}-\sqrt{\text{x}^2+\text{b}^2}\big)}{\big(\sqrt{\text{x}^2+\text{a}^2}-\sqrt{\text{x}^2+\text{b}^2}\big)}\Bigg]$
$=\lim\limits_{\text{x}\rightarrow\infty}\Bigg[\frac{\big(\sqrt{\text{x}^2+\text{a}^2}-\sqrt{\text{x}^2+\text{b}^2}\big)\big(\sqrt{\text{x}^2+\text{a}^2}-\sqrt{\text{x}^2+\text{b}^2}\big)\big(\sqrt{\text{x}^2+\text{c}^2}-\sqrt{\text{x}^2+\text{d}^2}\big)}{\big(\sqrt{\text{x}^2+\text{c}^2}-\sqrt{\text{x}^2+\text{d}^2}\big)\big(\sqrt{\text{x}^2+\text{c}^2}+\sqrt{\text{x}^2+\text{d}^2}\big)\big(\sqrt{\text{x}^2+\text{a}^2}+\sqrt{\text{x}^2+\text{b}^2}\big)}\Bigg]$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{\big(\text{x}^2+\text{a}^2\big)-\big(\text{x}^2+\text{b}^2\big)}{\big(\text{x}^2+\text{c}^2\big)-\big(\text{x}^2+\text{d}^2\big)}\times\bigg(\frac{\sqrt{\text{x}^2+\text{c}^2}+\sqrt{\text{x}^2+\text{d}^2}}{\sqrt{\text{x}^2+\text{a}^2}+\sqrt{\text{x}^2+\text{b}^2}}\bigg)$
$=\lim\limits_{\text{x}\rightarrow{\infty}}\Big(\frac{\text{a}^2-\text{b}^2}{\text{c}^2-\text{d}^2}\Big)\bigg(\frac{\sqrt{\text{x}^2+\text{c}^2}+\sqrt{\text{x}^2+\text{d}^2}}{\sqrt{\text{x}^2+\text{a}^2}+\sqrt{\text{x}^2+\text{b}^2}}\bigg)$
Dividing the numerator and the denominator by x:
$=\lim\limits_{\text{x}\rightarrow{\infty}}\Big(\frac{\text{a}^2-\text{b}^2}{\text{c}^2-\text{d}^2}\Big)\begin{pmatrix}\frac{\sqrt{1+\frac{\text{a}^2}{\text{x}^2}}+\sqrt{1+\frac{\text{a}^2}{\text{x}^2}}}{\sqrt{1+\frac{1}{\text{a}^2}}+\sqrt{1+\frac{\text{b}^2}{\text{a}^2}}}\end{pmatrix}$
$\text{As x}\rightarrow\infty,\frac{1}{\text{x}},\frac{1}{\text{x}^2}\rightarrow0$
$=\Big(\frac{\text{a}^2-\text{b}^2}{\text{c}^2-\text{d}^2}\Big)\Big(\frac{\sqrt{1}+\sqrt{1}}{\sqrt{1}+\sqrt{1}}\Big)$
$=\frac{\text{a}^2-\text{b}^2}{\text{c}^2-\text{d}^2}$

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