Question

Examine the continuity of f , where f is defined by

Answer 1

The given function is,

f( x )={ sinx−cosx,    if  x≠0 −1,                 if  x=0

The left hand limit of the function is,

lim x→ 0 − f( x )= lim x→ 0 − ( sinx−cosx ) = lim x→0−h ( sinx−cosx ) = lim h→0 ( sin( −h )−cos( −h ) ) = lim h→0 ( −sinh−cosh )

Solve for the left hand limit.

lim x→ 0 − f( x )=0−1 =−1

The right hand limit of the function is,

lim x→ 0 + f( x )= lim x→ 0 + ( sinx−cosx ) = lim x→0+h ( sinx−cosx ) = lim h→0 ( sinh−cosh ) =sin0−cos0

Solve for the right hand limit.

lim x→ 0 + f( x )=−1

The exact value of the function for x=0is,

f( x=0 )=−1

So, lim x→ 0 − f( x )= lim x→ 0 + f( x )=f( x=0 )

It is observed that the condition of continuity of the function f at x=0 is fulfilled.

Hence, function f is a continuous function.