Limits, derivatives and integration
A limit describes nearby function values and may exist even when the function is undefined at the point. Factoring can reveal a removable discontinuity, but cancel a factor only away from the excluded input. Continuity requires the function value and the limit to agree.
A derivative describes instantaneous rate and tangent slope. Apply the power, product and chain rules to the structure actually present, including the inner derivative of a composition. A stationary point is a candidate extremum, not proof of a maximum or minimum. Check derivative signs, and include interval endpoints when comparing values on a closed interval. A model's physical domain may exclude some algebraic critical points.
Definite integration records signed accumulation. Evaluate an antiderivative at the upper bound and subtract its value at the lower bound. If geometric area is requested and the curve changes sign, split the interval and account for the signs. Recovering a function from its derivative also requires an initial condition to determine the integration constant. Keep derivative units distinct from accumulated-quantity units.