Limit Laws II

Basic limit laws involving quotients, powers, and roots help simplify complex expressions as variables approach specific values.

The Quotient Law states that the limit of a quotient equals the quotient of the limits, provided the denominator's limit exists and is not zero.

The Power Law states that if a function is raised to a positive integer power, its limit is obtained by raising the limit of the base function to that same power.

The Root Law states that the nth root of a function approaches the nth root of its limit, as long as the limit exists and is non-negative when n is even.

These laws apply to real-world scenarios, such as population density in a growing city, defined as the ratio of population to land area. As both quantities change over time, their limits model long-term behavior.

The Quotient Law finds the limit of density from the limits of population and area, if the area’s limit isn’t zero.

If infrastructure demand grows with the square of density, the Power Law gives its limiting value.

If per-person energy needs depend on the square root of the limiting value of demand, the Root Law applies, provided it is non-negative.