Final answer:
The limit of sin(x)/x as x approaches infinity is 0 because sin(x) remains bounded between -1 and 1 while x grows without bound, making the ratio approach zero.
Step-by-step explanation:
As x approaches infinity, the limit of sin(x)/x can be determined by considering the values of sine function and the behavior of the denominator. The sine function oscillates between -1 and 1, which means sin(x) will always be within this range, regardless of how large x gets.
On the other hand, as x approaches infinity, x grows without bound. When a finite number, such as the output of sin(x), is divided by an infinitely large number, the result gets closer and closer to zero. Therefore, the limit of sin(x)/x as x approaches infinity is 0.
This is because no matter what value sin(x) takes, as x becomes larger and larger, the ratio sin(x)/x becomes smaller and smaller, approaching zero. This behavior of the function adheres to the rules of limits and the properties of the sine function in trigonometry. Hence, the correct answer is that the limit exists and is equal to 0.