How can I use the "angle addition formula" and the formula "limit of x approaching 0 of sin(x)" to show that the limit of x approaching 0 sin(a+x) = sin(a) for all a?
sin (a+x) = sin a cos x + sin x cos a
As x-> 0, cos x -> 1 and sin a -> 0
Therefore sin (a+x) approaches sin a
To show that the limit of x approaching 0 of sin(a+x) is equal to sin(a) for all a, we can utilize the angle addition formula and the limit of x approaching 0 of sin(x).
The angle addition formula states that sin(x+y) = sin(x)cos(y) + cos(x)sin(y).
Let's substitute y = a into the formula:
sin(a+x) = sin(x+a) = sin(x)cos(a) + cos(x)sin(a)
Now, we want to find the limit of this expression as x approaches 0.
We know that the limit of x approaching 0 of sin(x) is 0. Therefore, we can replace sin(x) with 0 in the equation:
lim(x->0) (sin(x)cos(a) + cos(x)sin(a))
= lim(x->0) (0cos(a) + cos(x)sin(a))
= 0cos(a) + cos(0)sin(a)
= 0 + 1 * sin(a)
= sin(a)
Hence, we have shown that the limit of x approaching 0 of sin(a+x) is equal to sin(a) for all a using the angle addition formula and the limit of x approaching 0 of sin(x).