Question

lim (Vx2 -2% -1-x2 – 7x-3)

JEE/Engineering Exams

Maths

Solution

154

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( lim _{x rightarrow-infty}(sqrt{x^{2}-2 x-1}-sqrt{x^{2}-7 x-3}) ) Rationalising ( lim _{x rightarrow-infty}left(frac{sqrt{x^{2}-2 x-1}-sqrt{x^{2}-7 x-3}}{sqrt{x^{2}-2 x-1}+sqrt{x^{2}-7 x-3}}right)(sqrt{x^{2}-2 x-1}+sqrt{x^{2}-7 x-3}) ) ( lim _{x rightarrow-infty} frac{left(x^{2}-2 x-1right)-left(x^{2}-7 x-3right)}{sqrt{x^{2}-2 x-1}+sqrt{x^{2}-7 x-3}} ) ( lim _{x rightarrow-infty} frac{-2 x-1+7 x+3}{sqrt{x^{2}-2 x+1}+sqrt{x^{2}-7 x-3}} ) ( =lim _{x rightarrow infty} frac{5 x+2}{sqrt{x^{2}-2 x-1}+sqrt{x^{2}-7 x-3}} ) ( =lim _{x rightarrow-infty} frac{x(5+2 / x)}{x[sqrt{1-2 / x-1 / x^{2}}+sqrt{1-7 / x-3 / x^{2}}]} ) ( =frac{5+0}{sqrt{1-0}+sqrt{1-0}} ) ( =frac{5}{1+1}=frac{5}{2} in ) of nonder