Question

Examine the function

f(x) `{:(= x^2 cos (1/x)",", "for"  x ≠ 0),(= 0",", "for"  x = 0):}`

for continuity and differentiability at x = 0

Answer 1

f'(0) = `lim_("h" -> 0) ("f"(0 + "h") - "f"(0))/"h"`

= `lim_("h" -> 0) ("f"("h") - 0)/"h"`

= `lim_("h" -> 0) ("h"^2 cos (1/"h"))/"h"`

= `lim_("h" -> 0) "h" cos(1/"h")    ...[(because "h" -> 0),(therefore "h" ≠ 0)]`

= `[lim_("h" -> 0) "h"] xx [lim_("h" -> 0) cos(1/"h")]`

= [0] × [FInite number between – 1 and 1]

= 0

∴ f is differentiable at x = 0 and hence continuous at x = 0.