Let the function f(x) = sin[x], where [·] is the greatest integer function and g(x) = |x|. What is lim(x→0) f(x)/g(x) equal to?

For x → 0⁺: [x] = 0, so f(x) = 0, and f(x)/g(x) = 0/x = 0. For x → 0⁻: [x] = −1, so f(x) = sin(−1) = −sin 1, and g(x) = |x| → 0⁺, so f(x)/g(x) = −sin 1 / |x| → −∞. The left and right limits differ, so the limit does not exist.