While proving the continuity of function f(x)=sin(x) or f(x)=cos(x) why do we put sin(c+h) or cos(c+h) in place of x ? Please explain

Question

Example: Lim f(x) = Lim [sin x] = Lim [sin (c+h)]

X tends to 0

Answer 1

Because to prove continuity of sin (x) at x = c, we need to look at the behaviour of the function close to the point. So we choose small values of h, and consider the function sin ( c+h).

The definition of continuity is :

f(x) is continuous at x = c if and only if

for any a>0 we can find a value of h so that if ⌈ x - c⌉ < h, then ( f(x) - f(c) ⌈ < a.

In other words we can find an x, so that f(x) is as close to f(c) as we like, (0.01 units or 0.000000001 units) ie there are no 'holes' in our function around x=c.

Mike