The limit of sin(x) as x approaches infinity does not exist because the sine function oscillates between -1 and 1. As x increases, sin(x) continues to go up and down in this range without settling at any particular value.
To understand this better, we can look at the nature of the sine function. The sine function is periodic with a period of 2π, meaning it repeats its values every 2π units. Therefore, for any value of x, you can find values of x that are arbitrarily large where sin(x) will take on any value between -1 and 1 infinitely often.
Because the function never approaches a single number as x goes to infinity, we say that the limit does not exist. In mathematical terms, we express this by stating:
lim (x → ∞) sin(x) does not exist.