Applying Squeeze Theorem to sin(n)/n

Real Analysis · Axiom Academy

EXAMPLE Applying the Squeeze Theorem to Trapping a bounded-numerator sequence between two vanishing envelopes to evaluate Evaluate the limit of the sequence as . The numerator oscillates forever and never settles, so no limit law applies directly — but is bounded , and that is exactly the hook the Squeeze Theorem needs. The dots are the terms . They stay pinned between the two envelopes , which both close in on the axis — so the dots are squeezed to 0 . The numerator never settled, yet the sequence converged — because we controlled it from both sides . Here is the pattern worth keeping: Bound the wild part: has no limit, but holds for every n . Build the sandwich: dividing by n>0 keeps the direction and gives . Match the outer limits: both envelopes satisfy . Collect the prize: the trapped sequence is forced to the same value, so . If for all n beyond some index, and , then . Any time a factor is bounded but doesn't converge — , , (-1)^n — look for a (something ) envelope and let the squeeze finish the job.

This is the written version of the interactive lesson above. See the full Real Analysis course.