Calculus I · Limits and Continuity · lesson
Geometric Proof of the sin(x)/x Limit
Learn Geometric Proof of the sin(x)/x Limit with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style pract
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 Geometric Proof of the sin(x)/x Limit; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Why sin(x)/x Approaches 1 to Scaled Sine and Tangent Limits. Read the explanation first, predict each example’s next move, and only then compare the written solution.
The unit-circle comparison
Take an angle with
In a unit circle:
• the area of the inner triangle is ; • the area of the circular sector is ; • the area of the outer tangent triangle is .
Therefore,
Multiply by :
Since , divide by :
Taking positive reciprocals reverses the inequalities:
As , both outer expressions approach . Hence
The function is even because
Thus the left-hand limit is also , and the two-sided limit is .
For a unit circle and , the inner triangle lies inside the sector, which lies inside the outer tangent triangle.
Near zero, the arc length and the vertical height become nearly indistinguishable. Their ratio approaches . The unit-circle proof turns "nearly" into inequalities strong enough for the Squeeze Theorem.
The graph of has a removable hole at and approaches from both sides.
Use the exact pattern
Evaluate
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This is the fundamental trigonometric limit itself:
No algebra is needed. Recognizing the pattern is the method.
trig-flow-01Evaluate .
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Multiply and divide by .
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Original instruction with traceable references.
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