Question

For a real number x, let [x] denote the greatest integer less than or equal to x-

hen f(x) = $\frac{e^{1/x} - e^{- 1/x}}{e^{1/x} + e^{- 1/x}}$is-

• A. Continuous at some x
• B. Continuous at all x but f '(x) does not exist for some x
• C. f '(x) exists for all x but f '' (x) does not exist
• D. f ' (x) exists for all x

Answer 1

Answer: (D)

f ' (x) exists for all x

Answer 2

Q [x –π] = Integer

∴ tan π [x – π] = 0

∴  $\frac { \tan \pi [ x - \pi ] } { 1 + [ x ] ^ { 2 } }$ = 0 → Constant fn ⇒ always continuous and differentiable

⇒ f (x) , f ' (x) , f " (x) ..... = 0 ∀ x ∈ R