Calculus I · Limits and Continuity · lesson
Scaled Sine and Tangent Limits
Learn Scaled Sine and Tangent Limits with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.
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 Scaled Sine and Tangent Limits; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Geometric Proof of the sin(x)/x Limit to Limits of Ratios of Sine Functions. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Learning objectives
Rewrite trigonometric limits so that a factor has the standard form or .
Scaled Sine and Tangent Limits
The substitution pattern
If , then whenever . We want to create the denominator that matches the sine argument.
\(\sin(5x)/x\)
Evaluate
Show worked solution
The sine argument is , but the denominator is only . Multiply and divide by :
As , , so
Therefore,
Keep the outside constant
Evaluate
Show worked solution
Create in the denominator:
The standard factor approaches , so
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.