To find the limit of the expression y sin x as x approaches infinity, we need to analyze the behavior of the sine function and how it interacts with the variable y.
The sine function, sin x, oscillates between -1 and 1 for all real numbers x. This means that no matter how large x gets, sin x will never settle at a single value; instead, it will keep fluctuating.
Now let’s consider the expression y sin x. If we assume y is a constant (not depending on x), the product y sin x will also oscillate between -|y| and |y| as x approaches infinity. Therefore, y sin x does not approach a specific limit but continues to oscillate indefinitely.
In mathematical terms, we can conclude that:
- If y = 0, then y sin x = 0 for all x, and the limit is 0.
- If y ≠ 0, the limit does not exist due to the oscillating nature of the sine function.
Therefore, the limit of y sin x as x approaches infinity is:
0, if y = 0; does not exist, if y ≠ 0.