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When evaluating the limit of a partial sum $S_n$ expressed as a quotient of polynomials in $n$ as $n \to \infty$, dividing both the numerator and denominator by the highest power of $n$ in the denominator (specifically $n^2$) isolates the asymptotic behavior of each term. This transformation converts lower-order terms into fractions with $n$ in the denominator, which clearly tend to 0, thereby revealing that the numerator dominates and the limit is infinity.

Conditions: The expression is a quotient of polynomials in $n$.; The limit is taken as $n \to \infty$.; The highest power of $n$ in the denominator is $n^2$.

The units of the area moment of inertia are derived directly from its defining integral, $I = \int r^2 dA$. The differential area element $dA$ has units of length squared (e.g., $m^2$).

Conditions: Lengths are measured in meters in the specific example.; The integral represents an area moment of inertia.

Standard matrix addition and subtraction are defined for matching dimensions, because every position requires a corresponding entry. Different dimensions leave unmatched positions under this standard rule; a separately specified new convention would be a different operation.

Conditions: Matrices have different dimensions (e.g., one is 3x2 and the other is 2x2).

When computing the composition $A \circ B$, the matrix B represents the first transformation. Its columns are the images of the standard basis vectors under B.

Conditions: A and B are 3x3 matrices representing linear transformations.; The goal is to find the matrix representation of the composite transformation.