_ x (2 x+1)⁵⁰+(2 x+2)⁵⁰+(2 x+3)⁵⁰+ +(2 x+100)⁵⁰ (2 x)⁵⁰+(10)⁵⁰ =

Step-by-step solution

The limit _ x (2 x+1)⁵⁰+(2 x+2)⁵⁰+ +(2 x+100)⁵⁰ (2 x)⁵⁰+(10)⁵⁰ is evaluated by identifying dominant terms as x approaches infinity. Factor (2x)⁵⁰ from each term in the numerator: (2x+k)⁵⁰ = (2x)⁵⁰ (1 + k 2x )⁵⁰ for k = 1 to 100 . The denominator simplifies to (2x)⁵⁰ (1 + 10⁵⁰ (2x)⁵⁰ ) , with (2x)⁵⁰ dominant as x . Dividing numerator and denominator by (2x)⁵⁰ yields _ k=1 ¹⁰⁰ (1 + k 2x )⁵⁰ 1 + 10⁵⁰ (2x)⁵⁰ . As x , each (1 + k 2x )⁵⁰ 1 and 10⁵⁰ (2x)⁵⁰ 0 , so the limit simplifies to 100 1 1 = 100 . Answer: 100