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math / Calculus / free

Limits & Continuity

Approach a point from either side and see when the values agree.

Follow the mechanism1 / 3

Inputs move toward a target. A limit studies nearby behavior without requiring the input to equal the target.

approaching a pointcontinuous
Focus the idea
A function value approaches its limit from either sideFor y equals x squared, x approaches 2 from the right with a gap of 1.40. The nearby value is 11.56 and approaches 4. At x equals 2, the function is 4.x = 3.40limit = 4xf(x)
approach x3.40
nearby f(x)11.56
limit4.00
at x = 24.0
Approach from
At x = 2

The right-hand value is 11.56; shrink the gap to test what it approaches.

A limit describes the value a function approaches. Continuity at x = 2 means the function is defined there and both one-sided approaches agree with f(2).