_ x 5 2-2 (x^2-12 x+35 ) (x-5) =

Step-by-step solution

The limit L is evaluated as x 5 for 2 - 2 (x^2 - 12x + 35) x - 5 . Using the identity 1 - A = 2 ^2 ( A 2 ) , the numerator simplifies to 4 ^2 ( x^2 - 12x + 35 2 ) = 2 | ( (x-5)(x-7) 2 ) | . For the left-hand limit as x 5^- , note that (x-5) 0 . Thus the expression inside the sine is positive, and the absolute value is unnecessary: 2 ( (x-5)(x-7) 2 ) . Applying the standard limit _ 0 = 1 : _ x 5^- 2 ( (x-5)(x-7) 2 ) x-5 = _ x 5^- (x-7) = -2 Since the right-hand limit yields a different value, the two-sided limit doe