AP® Calculus AB · Unit 1 · Topic 1.8
The Special Trigonometric Limits
As x approaches 0, sin(x)/x approaches 1 and (1 - cos(x))/x approaches 0. These two special limits let you crack otherwise unsolvable trig limits by matching arguments and rescaling.
Plug x = 0 into sin(x)/x and you get 0/0, undefined, yet the graph shows the function closing in on 1 from both sides. Picture a tiny angle x on the unit circle: the arc length is x, and the vertical sine segment is sin(x). As the angle shrinks, that straight sine segment and the curved arc it cuts off become almost indistinguishable in length, so their ratio sin(x)/x squeezes toward 1. The companion limit, (1 - cos(x))/x approaching 0, comes from the same picture: cos(x) hugs 1 so tightly near x = 0 that 1 - cos(x) shrinks faster than x itself.
The catch is that the basic limit only works when the expression inside sine matches the denominator exactly: sin(u)/u needs that same u underneath. When it doesn't match, rescale. For sin(5x)/(3x), rewrite it as (5/3) times sin(5x)/(5x), so the inner argument 5x lines up with the denominator 5x, and the whole thing approaches (5/3)(1) = 5/3. The same move handles (1-cos(x))/x variants once you recognize the family and adjust the denominator's power and the argument together.
These limits matter because they are the gateway to differentiating sine and cosine from first principles: the derivative of sin(x) falls straight out of sin(x)/x approaching 1. They also round out the limit toolkit built in this unit, sitting alongside direct substitution, factoring, and algebraic cancellation as another technique for resolving 0/0 forms, this time ones that algebra alone cannot touch.
Key terms
- Special trigonometric limit
- The fact that sin(x)/x approaches 1 and (1-cos(x))/x approaches 0 as x approaches 0.
- Indeterminate form
- An expression like 0/0 that gives no information about a limit until it is rewritten.
- Unit circle
- A circle of radius 1 centered at the origin, used to define sine and cosine as coordinates of a point.
- Rescaling
- Multiplying and dividing by a constant so the argument inside a trig function matches its denominator.
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End of topic.