DAY 2002A Limits Practice Test

When the numerator stays bounded while the denominator grows without bound, the fraction must go to zero. The sine function is always between -1 and 1, so for all x we have |sin x| ≤ 1. This gives |sin x / x| ≤ 1/|x|. As x approaches infinity, 1/|x| tends to 0, so the whole expression sin x / x must approach 0 as well. The oscillations of sin x fade away because they’re being divided by an unboundedly large number. Hence the limit is 0.