If _ x 0 ^2 x ( 3 ^2 x + 4 x + 9 - 2 ^2 x + a x + b ) = 1 4 , where a and b are real constants, then the value of a - b is equal to

Step-by-step solution

For the limit to exist and be finite, the numerator must approach 0 as x 0 , because the denominator ^2 x approaches 0 . Substituting x = 0 into the numerator gives: 3(1) + 4(1) + 9 - 2(1) + a(1) + b = 0 16 = 2 + a + b a + b = 14 Now, rationalizing the numerator: _ x 0 3 ^2 x + 4 x + 9 - (2 ^2 x + a x + b) ^2 x ( 3 ^2 x + 4 x + 9 + 2 ^2 x + a x + b ) _ x 0 ^2 x + (4 - a) x + (9 - b) (1 - ^2 x) ( 16 + 16 ) Since a + b = 14 , the quadratic in the numerator is ^2 x + (4 - a) x + (9 - b) . We know x = 1 is a root, so i