If _ x (1+ a x + b x^2 )^ 2 x =e^2 , then

Step-by-step solution

The given limit is of the form 1^ . We use the formula _ x (f(x))^ g(x) = e^ _ x g(x)(f(x)-1) . Here, f(x) = 1 + a x + b x^2 and g(x) = 2x . The limit becomes e^ _ x 2x (1 + a x + b x^2 - 1 ) = e^ _ x 2x ( a x + b x^2 ) . Simplifying the expression inside the exponent: _ x 2x ( ax + b x^2 ) = _ x 2ax^2 + 2bx x^2 = _ x (2a + 2b x ) = 2a . We are given that the limit is e^2 , so e^ 2a = e^2 . Equating the exponents, 2a = 2 , which gives a = 1 . The value of b does not affect the limit as _ x 2b x = 0 for any finite b