find a non-zero value for the constant "k" so that

f(x) =tan(kx)/x, if x < 0
3x + 2k^2, if x> 0
will be continuous at x=0

To find a non-zero value for the constant "k" such that the function f(x) is continuous at x = 0, we need to ensure that the left-hand limit and the right-hand limit of the function are equal at x = 0.

To find the left-hand limit, we need to evaluate the function f(x) as x approaches 0 from the left side (x < 0). In this case, the function is defined as f(x) = tan(kx)/x.

Taking the limit as x approaches 0 from the left:

lim(x→0-) [tan(kx)/x]

We know that the limit of tan(x)/x as x approaches 0 is equal to 1. Therefore, we have:

lim(x→0-) tan(kx)/x = 1

Next, we find the right-hand limit as x approaches 0 (x > 0). In this case, the function is defined as f(x) = 3x + 2k^2.

Taking the limit as x approaches 0 from the right:

lim(x→0+) 3x + 2k^2 = 2k^2

For the function to be continuous at x = 0, the left-hand limit and the right-hand limit must be equal. So we have:

1 = 2k^2

Solving for k:

2k^2 = 1
k^2 = 1/2
k = ±√(1/2)

Therefore, we have two possible non-zero values for the constant k that make the function continuous at x = 0:

k = √(1/2) or k = -√(1/2)