Question

Show that the function f(x) = 2x - |x| is continuous at x = 0

Answer 1

`"L" ["f"(x)]_(x=0) = lim_(x->0^-) "f"(x) = lim_(h->0) "f"(0-"h")`

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

`= lim_(h->0) 2 (-"h") - |"h"|`

`= lim_(h->0) - 2"h" - "h"`

`= lim_(h->0) -3"h"`

= - 3(0) = 0   ....(1)

`"R" ["f"(x)]_(x=0) = lim_(x->0^+) "f"(x) = lim_(h->0^+) "f"(0 + "h")`

= `lim_(h->0) "f"("h") = lim_(h->0) 2"h" - |"h"|`

= `lim_(h->0) 2"h" - "h"`

= `lim_(h->0) "h" = 0`   ...(2)

Also f(0) = 2(0) - |0| = 0

From (1), (2) and (3),

`"L" ["f"(x)]_(x=0) = "R" ["f"(x)]_(x=0)` = f(0) = 0

∴ f(x) = 2x - |x| is continuous at x = 0.