Squeeze theorem

The squeeze theorem states that a function converges to a limit if it lies between two functions that converge to that same limit.

Limit of a function: A value that a function’s outputs approach as its input approaches a specified point. The theorem establishes this kind of limit by trapping the function between two bounds.
Trigonometric limits: Limits involving trigonometric functions as their inputs approach specified values. Geometric bounds yield the fundamental limit of sine divided by its angle.
Continuity: A property of a function whose values change without jumps at a point or throughout a domain. Continuity supplies many outer functions whose limits are easy to calculate.
L'Hôpital's rule: A method for evaluating certain indeterminate limits by differentiating numerator and denominator. It can evaluate ratios directly, while squeezing proves limits through inequalities.
Order relation: A relation that compares elements as less than, equal to, or greater than one another. The pointwise inequalities are what let the outer limits constrain the middle function.
Limit of sin x / x: The limit of sin x divided by x is 1 as x approaches zero, with angles measured in radians. This standard result follows by squeezing the ratio between cosine and 1.
Monotonic function: A function that consistently increases or consistently decreases over an interval. Monotonicity can help construct or verify upper and lower bounds.
Sandwich theorem for integrals: A result bounding an integral between integrals of pointwise lower and upper functions. It resembles squeezing but concerns integral values rather than pointwise function limits.
Convergence: The approach of a sequence or function toward a specified value or state. The theorem is a tool for proving convergence from bounds.
Cosine: A trigonometric function that gives the horizontal coordinate of a point on the unit circle. Near zero, cosine provides one side of the bounds used in the sine-limit proof.

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