Calculus I · Limits and Continuity · lesson
Cosine Limits and Trigonometric Identities
Learn Cosine Limits and Trigonometric Identities with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style
Where this chapter fits
Chapter 3: Trigonometric limits
Use squeezing, the fundamental sine limit, identities, and scaling to make trigonometric behavior predictable.
Reading lens: Can the expression be rewritten around a known small-angle limit, with every scaling factor accounted for? Keep that question in view while reading Cosine Limits and Trigonometric Identities; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Limits of Ratios of Sine Functions to Trigonometric Limit Decision Guide. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Use conjugates and identities to reduce limits involving , , or other trigonometric expressions to standard forms.
Cosine Limits and Trigonometric Identities
A first cosine limit
To show this, rationalize with :
The first factor approaches , and the second approaches . Therefore the product approaches .
The second important cosine limit
Again rationalize:
Taking limits gives
Recognize the squared pattern
Evaluate
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Rewrite as a square:
The inside approaches , so the square approaches
Identity plus a standard limit
Evaluate
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Create in the denominator:
As , , so the standard cosine limit gives
Exam-level: use an identity before the standard limit
Evaluate
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Separate the two standard factors:
Each factor can be rewritten:
Therefore the product approaches
Do not replace by as an algebraic identity. The statement near zero means their ratio approaches ; it does not mean the two expressions are equal for nonzero . Preserve the limit argument.
Source & rights
Original instruction with traceable references.
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