Find the values of a and b such that the function f(x) is continuous at x = 0

Question

f(x) =

(sin(ax))(sin(1/x)) if x ≠ 0

b if x =0

Answer 1

In order for a function to be continuous at a point (like x=0), the limits of the function as it approaches the point from the left and the right must be equal. In other words:

lim_x->0^- f(x) = lim_x->0⁺ f(x)

In this case, the function is constant to the right of x=0, so the limit from the right is easy. The limit from the left is not so obvious, but it looks like the squeeze theorem will be helpful. We know how sin(ax) and -sin(ax) will behave as x->0, and we know that -1 ≤ sin(1/x) ≤ 1 for all x ≠ 0. Thus, -sin(ax) ≤ sin(ax)sin(1/x) ≤ sin(ax). That should get you to an answer. I'll be checking in if you have any questions.