Question

The function f(x) = x² – sin x + 5 is continuous at x =

Options

• `pi/6`

• `pi/4`

• `pi/2`

• `pi`

Answer 1

`pi`

Explanation:

Let `f(x) = x^2 - sin x + 5`

L.H.L = `lim_(x -> pi^-) (x^2 - sin x + 5)` put `x = pi - h`

= `lim_(h -> 0) [(pi - h)^2 - sin(pi - h) + 5]`

= `lim_(h -> 0) [pi^2 - 2pih + h^2 - sin h + 5] = pi^2 + 5`

R.H.L `lim_(x -> pi^+) [pi^2 - sin x + 5]`, put `x = pi + h`

= `lim_(h -> 0) [(pi + h)^2 - sin(pi + h) + 5]`

= `lim_(h -> 0) [pi^2 - 2pih + h^2 + sin h + 5] = pi^2 + 5`

`f(pi) = pi^2 + 5`

∴ L.H.L = R.H.L = `f(pi)`

Hence `f` is continuous at `x = pi`.