Question

plz plz help me for this!!!

Answer 1

Answer:

A function f(x) is continuous at x=a if:

• f(a) is defined
• lim f(x)  [ as x->a ] exists
• the two values above are equal

For the given function:

(a) f(0) is defined.  It is specified that f(0) = -1.

(b) As sin x and cos x are continuous, lim f(x) [ as x->0 ] exists.  The value of the limit is:

  lim f(x) = lim ( sin x - cos x ) = lim sin x - lim cos x  [ all limits as x->0 ]

             = sin 0 - cos 0    [ since sin and cos are continuous ]

             = 0 - 1

             = -1

(c) Since lim f(x) [ as x->0 ]  =  f(0)       [ both are equal to -1 ]

the function f(x) is continuous at x = 0.