Question

See the attachment and do help me fast.

Answer 1

Lim(x→1) { -ax + sin(x -1) + a }/{x + sin(x -1) -1}^(1-x)/(1-√x) = 1/4

we check Limit after putting x = 1
Limit is in the form of 0/0 below and above
so,

Lim(x→1) { -ax + sin(x -1) + a }/{x + sin(x -1) -1}^(1-√x)(1+√x)/(1-√x) =1/4

use L - Hospital rule

Lim(x→1) [{ -a + cos(x -1) }/{1 + cos(x -1)}]² = 1/4

{(-a +1)/(2)}² = 1/4

(-a +1)²/4 = 1/4

(-a +1)² = 1

-a +1 = ±1
a = 0 or 2