Limit laws and legitimate algebra

Try

Evaluate (x²−9)/(x−3) at x=2.9 and x=3.1. The results are 5.9 and 6.1. At x=3 the formula is undefined. Explain how a limit might still be 6 without assigning a value to the original fraction.

Predict

A learner says, “Both numerator and denominator approach zero, so their quotient approaches zero.” Test this claim against x/x, x²/x and x/x² near zero. Predict whether the three expressions can have the same behavior.

Experiment

Factor before using a calculator:

Expression away from the excluded point	Equivalent nearby expression	Proposed limit
(x²−9)/(x−3), x approaching 3	x+3	6
x²/x, x approaching 0	x	0
x/x, x approaching 0	1	1
x/x², x approaching 0	1/x	no finite two-sided limit

Each cancellation assumes the canceled factor is nonzero. Circle that restriction in your work. The restriction is compatible with a limit, which does not require evaluating at the approached point.

Understand

If f(x) approaches A and g(x) approaches B, their sum approaches A+B and their product approaches AB. Multiplication by a fixed constant also passes through a limit. The quotient approaches A/B provided B is nonzero. That condition is part of the rule, not a minor exception.

Polynomials are continuous, so their limits at finite inputs can be evaluated by substitution. Rational expressions can be evaluated this way where their denominator is nonzero. When substitution produces 0/0, it has produced an undefined expression, often called an indeterminate form. It has not calculated the limit. The examples above show why further work is necessary.

For a square-root difference, use a conjugate. Consider (√(x+9)−3)/x near zero. Multiply numerator and denominator by √(x+9)+3. The numerator becomes x+9−9=x. For x≠0, cancel it to obtain 1/(√(x+9)+3). The new denominator approaches 6, so the limit is 1/6. The real square root exists throughout a neighborhood of zero, and the conjugate is nonzero there; those checks justify the manipulation.

This method relies on equality at nearby allowed inputs. It does not claim that the original and simplified formulas have identical domains. Similarly, knowing that a denominator limit is zero does not automatically prove an infinite limit: signs, numerator behavior and one-sided approaches matter.

Apply

Use this decision sequence in the practice bank: identify the domain near the target; try permitted substitution; if it is indeterminate, choose an algebraic identity; state where the identity holds; then apply a limit law whose hypotheses are satisfied. For the error-repair problem, explicitly explain the invalid step as well as giving the corrected number.

Continue

Some expressions cannot be simplified into an easily evaluated formula. Bounds can still establish their limit. The next lesson handles oscillating functions by controlling their distance from a target rather than tracing every oscillation.