Question

The value of `lim_{x→0} (1 + sinx - cosx + log_e(1 - x))/x^3` is ______

विकल्प

• -1

• `1/2`

• `-1/2`

• 1

Answer 1

The value of `lim_{x→0} (1 + sinx - cosx + log_e(1 - x))/x^3` is `underline(-1/2)`.

Explanation:

`lim_{x→0} (1 + sinx - cosx + log_e(1 - x))/x^3` 

`= lim_{x→0} 1/x^3[1 + (x - x^3/(3!) + x^5/(5!) - x^7/(7!) + .......) - (1 - x^2/(2!) + x^4/(4!) - x^6/(6!) + ......) - (x + x^2/2 + x^3/3 + x^4/4 + .....)]`

= `lim_{x→0} (-x^3(1/(3!) + 1/3) - x^4(1/(4!) + 1/4) + ....)/(x^3)`

= `lim_{x→0} [-(1/(3!) + 1/3) - x(1/(4!) + 1/4) + .....]`

= `-1/2`