LimitsMedium

Limit of sin(x)/x as x Approaches 0

Evaluate the fundamental trigonometric limit and understand why direct substitution is inconclusive.

lim x→0 sin(x)/x

Method: Fundamental trigonometric limit

Step-by-step solution

1
Try substitution
Substituting x=0 produces 0/0, an indeterminate form rather than an answer.
2
Use the standard limit
For angles measured in radians, the sine ratio approaches one.
$lim_{x o0} rac{sin x}{x}=1$
3
Check both sides
The left-hand and right-hand values approach the same number.

1

A geometric squeeze argument or a local series expansion establishes this limit; it is also the basis for differentiating sine.

Treating 0/0 as zero skips the limiting process and gives the wrong conclusion.

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