A Trigonometric limit at 0 (Calculus)

Added: 2026-05-06

[00:00:00]We are examining here a very interesting limit here at 0 1 - cosine x / sin^ square x. Think of this fundamental identity sin square of a variable plus cosine square of that same variable is always equal to 1. When you solve for sin square is equal to 1 - cosine square. You see this and you'll substitute this right over here.

[00:00:18]Suddenly your limit will modify into this. You will have here 1 - cosine x in the numerator and then you'll have 1 - cosine^ square x in the denominator. And you can separate this into its factors.

[00:00:28]When you do it, you have limit here as x approaches 0 1 - cosine x / 1 - cosine x * 1 + cosine x. This thing here is like a a square - b square. Then you have a - b * a + b. You know, obviously this cancels out. I'm looking here now at limit as x approaches 0. We have 1 / 1 + cossine x. Use your direct substitution property of limits. 0 can come right over here. Cosine of 0 is a one. Your end result here is 1 / 1 + 1 which is 1 / 2. And this here is your end result and it will certainly be correct.