Question

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

Answer 1

We have, f(x) = `{{:(|x - "a"| sin  1/(x - "a")",",  "if"  x ≠ 0),(0",",  "if"  x = "a"):}` at x = a

 At x = a

L.H.L. = `lim_(x -> "a"^-) |x - "a"| sin  1/(x - "a")`

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

= `lim_("h" -> 0) - "h" sin  1/"h"`

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

R.H.L. = `lim_(x -> "a"^+) |x - "a"|sin(1/(x - "a"))`

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

= `lim_("h" -> 0) "h" sin  1/"h"`

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

Also f(a) = 0  ...(Given)

Thus L.H.L. = R.H.L. = f(a)

So, f(x) is continuous at x = a.