For what values of #x# is the function #f(x) = sin x + cos x# continuous?
1 Answer
It is continuous on the whole of
Explanation:
Given
If you are happy that
On the other hand, if you want to prove it from basic principles, we can proceed as follows:
We find:
#f(x) = sin x + cos x#
#color(white)(f(x)) = sqrt(2)(sqrt(2)/2 sin x + sqrt(2)/2 cos x)#
#color(white)(f(x)) = sqrt(2)(sin x cos (pi/4) + cos x sin (pi/4))#
#color(white)(f(x)) = sqrt(2)sin (x+pi/4)#
Then:
#f(x+delta) = sqrt(2)sin (x + pi/4 + delta)#
#color(white)(f(x+delta)) = sqrt(2)(sin (x + pi/4) cos delta + cos (x + pi/4) sin delta)#
Note that:
#{ (lim_(delta->0) sin delta = 0), (lim_(delta->0) cos delta = 1) :}#
So:
#lim_(delta->0) f(x + delta) = lim_(delta->0) sqrt(2)(sin (x + pi/4) cos delta + cos (x + pi/4) sin delta)#
#color(white)(lim_(delta->0) f(x + delta)) = sqrt(2)(sin (x + pi/4) (1) + cos (x + pi/4) (0))#
#color(white)(lim_(delta->0) f(x + delta)) = sqrt(2)sin (x + pi/4)#
#color(white)(lim_(delta->0) f(x + delta)) = f(x)#
So