Continuity With Piece Wise Functions

[Michele Nunes]

Homework Statement

Determine all values of the constant a such that the following function is continuous for all real numbers.
f(x) = ax/tan(x), x ≥ 0
= a² - 2, x < 0

Homework Equations

The Attempt at a Solution

I tried so many different ways to get the first part of the function to be defined at 0 but nothing worked, I tried manipulating it with a bunch of trig identities and no matter what, that first part is always undefined at 0 so I don't know how the function can ever be continuous if that first part of the function is always going to be undefined at 0 and I can't remove it.

 

[Mark44]

Homework Statement

Determine all values of the constant a such that the following function is continuous for all real numbers.
f(x) = ax/tan(x), x ≥ 0
= a² - 2, x < 0

Homework Equations

The Attempt at a Solution

I tried so many different ways to get the first part of the function to be defined at 0 but nothing worked, I tried manipulating it with a bunch of trig identities and no matter what, that first part is always undefined at 0 so I don't know how the function can ever be continuous if that first part of the function is always going to be undefined at 0 and I can't remove it.

This limit will be helpful:
$$\lim_{x \to 0} \frac {\sin x} x = 1$$

Note that $\frac{ax}{\tan(x)} = a \frac x {\frac{\sin(x)}{\cos(x)}}$