Calculus I · Limits and Continuity · lesson
Limits of Ratios of Sine Functions
Ratio of two sine expressions
Sine over sine
Evaluate
Show worked solution
Insert factors that create both standard limits:
The first and third factors approach , so
Tangent
Since
we have
A tangent limit
Evaluate
Show worked solution
Create in the denominator:
The standard tangent factor approaches , so
After the explanation
Use the section idea
Can the expression be rewritten around a known small-angle limit, with every scaling factor accounted for?
The fundamental sine limit is a reusable local shape: other trigonometric limits work when you expose that shape through identities and scaling.
Look for a bounded oscillation times a shrinking factor, or rewrite the expression into sine-over-angle factors whose arguments match their denominators.
The sine function is not equal to its angle; their ratio merely approaches one near zero, and that statement requires radian measure.
Mastery means you can mark every scaling factor before simplifying and can explain where the Squeeze Theorem enters the argument.
Source & rights
Original instruction with traceable references.
The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.
The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.