Calculus I · Limits and Continuity · lesson
Geometric Proof of the sin(x)/x Limit
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 .
Read this graph as text
Unit-circle squeeze geometry. A unit circle is centered at O = (0, 0). A = (1, 0), B = (cos theta, sin theta), and T = (1, tan theta). Segments form an inner triangle OAB and an outer tangent triangle OAT, while the circle arc and radii bound the sector AOB. Patterned regions and a written inequality show inner triangle area less than or equal to sector area less than or equal to outer triangle area.
The inner triangle has diagonal hatching, the sector has dots, and the outer triangle has crosshatching. Point shapes and line styles are distinct and all points are labeled.
Why it matters: Compare the areas of an inner triangle, circular sector, and outer tangent triangle to establish the inequalities behind the sine limit.
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.
Read this graph as text
The sine-over-x limit. The even function y = sin(x) divided by x is drawn from approximately negative 2 pi to 2 pi in two explicit branches that stop at zero. An open circle at (0, 1) marks the missing formula value. A dashed horizontal guide marks y = 1, and both branches approach that guide as x approaches zero.
The two solid branches stop at an open circle, and a dashed horizontal guide with text marks y = 1.
Why it matters: Show graphically that sin(x)/x approaches 1 from both sides even though the displayed formula is undefined at zero.
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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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.
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