does sin(1/x) converge or diverge?

for what values of x? Sine is always between 1 and -1

the series sin(1/x) from 0 to infinityy

To determine whether the function sin(1/x) converges or diverges as x approaches 0, we need to analyze its behavior as x approaches 0 from both the left and the right.

First, let's consider the limit as x approaches 0 from the right (x → 0⁺). In this case, we can approach 0 by taking x values that are positive and very close to zero. As x approaches 0 from the right, the function sin(1/x) oscillates infinitely between -1 and 1. Therefore, sin(1/x) does not converge as x approaches 0 from the right.

Next, let's examine the limit as x approaches 0 from the left (x → 0⁻). Here, we approach 0 with negative values of x very close to zero. Similar to the previous case, sin(1/x) oscillates infinitely between -1 and 1 as x approaches 0 from the left. Therefore, sin(1/x) does not converge as x approaches 0 from the left.

Since sin(1/x) does not converge as x approaches 0 from either direction, we can conclude that the function sin(1/x) diverges as x approaches 0.