How to bound trigonometric functions to find limits at infinity
In short: When a trig function is trapped in a quotient, finding the limit at infinity can be tricky. This video demonstrates how to use the Squeeze Theorem by bounding the numerator between two simpler rational functions.
The lesson focuses on evaluating the limit of cos(x)/(x-1) as x approaches infinity. Because trigonometric functions like cosine oscillate, they don't settle on a single value, making direct evaluation difficult. Instead, the method uses the fact that -1 and 1 are the bounds for cos(x).
By placing the original expression between two rational limits—specifically 1/(x-1) and -1/(x-1)—the problem becomes solvable. This technique illustrates a core principle of calculus: if you can trap a complex function between two simpler ones that both head toward zero, the middle function must also approach zero.
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Sources
- Summary by Intellect · source: Limits at infinity of quotients with trig | Limits and continuity | AP Calculus AB | Khan Academy
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