QJEEM21Ma1a45: JEE Main 2021 Mathematics - Class 12 JEEM21Ma1a45 Solution: If , then a + b + c is equal to ____________.

Question

Verified Solution • 2025–26 Session moderate

If

\mathop {\lim }\limits_{x \to 0} {{a{e^x} - b\cos x + c{e^{ - x}}} \over {x\sin x}} = 2

, then a + b + c is equal to ____________.

Accepted Answer

\mathop {\lim }\limits_{x \to 0} {{\left\{ {a\left( {1 + x + {{{x^2}} \over {2!}} + .....} \right) - b\left( {1 - {{{x^2}} \over {2!}} + {{{x^4}} \over {4!}}......} \right) + c\left( {1 - x + {{{x^2}} \over {2!}}......} \right)} \right\}} \over {x\left( {x - {{{x^3}} \over {3!}} + .....} \right)}} = 2

\therefore

\mathop {\lim }\limits_{x \to 0} {{(a - b + c) + x(a - c) + {x^2}\left( {{a \over 2} + {b \over 2} + {c \over 2}} \right) + ....} \over {{x^2}\left( {1 - {{{x^2}} \over 6}....} \right)}} = 2

For this limit to exist

a

-

b + c = 0 & a

-

c = 0

&

{a \over 2} + {b \over 2} + {c \over 2} = 2

\Rightarrow

a + b + c = 4

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