Question

It can also be solved by L'Hospital rule Find the value of a, b and c so that lim ae" - b cos x + cet -=2 xsin x Ex.8 10

SSC CGL

Maths

Solution

147

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( lim _{x rightarrow 0} frac{a e^{x}-b cos x+c e^{-x}}{x sin x}=2 ) at ( x=0 ) As linot exists and denuonimator - o suilth no numerator must tond to 0 At ( x=0 ) [ begin{array}{l} a e^{0}-b cos (0)+c e^{-0}=0 Rightarrow a-b+c=0 end{array} ] ( lim _{x rightarrow 0} frac{a e^{x}-b cos x+c e^{-x}}{x^{2}(sin x)} frac{x}{x} ) ( operatorname{asin} frac{sin x}{x} rightarrow 1 ) at ( x=0 ) it can me eliminatia ( =lim _{x rightarrow 0} frac{a e^{x}-b cos x+c e^{-x}}{x^{2}} ) Applying L Hopital's rule ( =lim _{x rightarrow 0} frac{a e^{x}+b sin x-c e^{-x}}{2 x} ) As demominator ( rightarrow 0 ) at ( x=0 ) [ 9 e^{0}+b ] ( 1-e^{e^{0}}=0 ) Using (i) and (11) ( frac{a e^{x}+2 a operatorname{sen} x-a e^{-x}}{x}=2 ) [ = ] ( =lambdaleft(frac{e^{x}-1}{x}+frac{2 a sin x+aleft(frac{-x}{1}-1right)}{x}=2right. ) ( 4 a=2 ) ( Rightarrow quad 4 a=2 quad b=1 quad c=1 / 2 )