_ x 0 (5^x - 1)^4 , cosec ,(x 5) (x 5) (1 + x^2 25) =

Step-by-step solution

The given limit is: _ x 0 (5^x - 1)^4 , cosec ,(x 5) (x 5) (1 + x^2 25) Rewriting cosec ,(x 5) as 1 (x 5) , we get: _ x 0 (5^x - 1)^4 (x 5) (x 5) (1 + x^2 25) Dividing the numerator and the denominator by x^4 : _ x 0 ( 5^x - 1 x )^4 ( (x 5) x ) ( (x 5) x ) ( (1 + x^2 25) x^2 ) Using the standard limits _ x 0 a^x - 1 x = a , _ x 0 (kx) x = k , _ x 0 (kx) x = k , and _ x 0 (1 + kx) x = k , we evaluate the limits of the individual terms: Numerator limit = ( 5)^4 Denominator limit = ( 5) ( 5) ( 25) Since 25 = (5^2) = 2