Question

The value of `lim_(x → ∞) ((x^2 - 1)sin^2(πx))/(x^4 - 2x^3 + 2x - 1)` is equal to ______.

Options

• `π^2/6`

• `π^2/3`

• `π^2/2`

• π²

Answer 1

The value of `lim_(x → ∞) ((x^2 - 1)sin^2(πx))/(x^4 - 2x^3 + 2x - 1)` is equal to `underlinebb(π^2)`.

Explanation:

Let A = `lim_(x → 1) ((x^2 - 1)sin^2(πx))/(x^4 - 2x^3 + 2x - 1) (0/0 "forms")`

⇒ A = `lim_(x → 1) ((x^2 - 1)sin^2πx)/((x^4 - 1) - (2x^3 - 2x))`

⇒ A = `lim_(x → 1) ((x^2 - 1)sin^2πx)/((x^2 - 1)(x^2 + 1) - 2x(x^2 - 1))`

⇒ A = `lim_(x → 1) ((x^2 - 1)sin^2πx)/((x^2 - 1)(x^2 + 1 - 2x))`

⇒ A = `lim_(x →1) (sin^2πx)/((x - 1)^2`

Put x = 1 + h

⇒ A = `lim_(h → 0) (sin^2π(1 + h))/h^2 = lim_(h → 0) (-sin πh)^2/h^2`

⇒ A = `lim_(x → 1) ((sin πh)/(πh))^2.π^2`

⇒ A = π²  ...`{∵  lim_(x → 0) sinx/x = 1}`