Statement -1: `lim_(xrarr pi//2 ) (cot x-cosx)/((2x-pi)^3)=-(1)/(16)` Statement 2 `lim_(xrarr0) (tanx-sinx)/(x^3)=(1)/(2)`

Options

A. Statement -1 is true, Statement-2 is true,, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is true, Statement-2 is true, Statement-2 is not a correct explanation for statement -1.
C. Statement-1 is true, Statement-2 is False.
D. Statement-1 is False, Statement-2 is true.

Answer

A

Explanation

Clearly,
`lim_(xto 0) (tanx-sinx)/(x^3) lim_(xto0) (sinx)/(x)xx(1-cos x)/(x^2)=1xx(1)/(2)=(1)/(2)`
So, statement -2 is true.
Now, `lim_(xto pi//2)(cot x-cos x)/(2x-pi^3)`
` =-(1)/(8)lim_(xtopi)(tan(pi//2-x)-sin(pi//2-x))/((pi//2-x)^3)`
`=(1)/(8)xx(1)/(2)=-(1)/(16)`
So, both the statements are true and statement -2 is a correct explanation for statement -1.