Question

f(x) = `{{:(|x|cos  1/x",", "if"  x ≠ 0),(0",", "if"  x = 0):}` at x = 0

Answer 1

We have,  `{{:(|x|cos  1/x",", "if"  x ≠ 0),(0",", "if"  x = 0):}`

At x = 0

L.H.L. = `lim_(x -> 0^-) |x| cos  1/x`

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

= `lim_("h" -> 0) "h" cos  1/"h"`

= 0 × [an oscillating number between –1 and 1] = 0

R.H.L. = `lim_(x -> 0^+) |x| cos  1/x`

= `lim_("h" -> 0) |0 + "h"| cos  1/(0 + "h")`

= `lim_("h" -> 0) "h" cos  1/"h"`

= 0 × [an oscillating number between –1 and 1] = 0

Also f(0) = 0  ....(Given)

Thus, L.H.L. = R.H.L. = f(0)

So, f(x) is continuous at x = 0