Apply the limit laws — sum, difference, product, quotient, constant multiple, power, and root — to evaluate limits step by step by breaking composite expressions into simpler pieces.
+ - * / ** Math.sqrt() Math.abs() Math.sin() Math.cos() Math.log() Math.exp()| Law | Expression | Result |
|---|
lim(x→a)[f(x) ★ g(x)] = L ★ M
The limit laws are valid whenever the individual limits exist (are finite real numbers). Specifically:
Continuity connection: If f is continuous at a, then lim(x→a) f(x) = f(a). Polynomials are continuous everywhere, so you can always substitute directly.
Quotient caveat: The Quotient Law requires lim g(x) = M ≠ 0. If M = 0, the law breaks down and you need to investigate further.
Root caveat: For even-index roots, the limit L must be ≥ 0 so that the root is a real number.
0/0 ∞/∞ ∞ − ∞ 0·∞
0/0 form: The most common indeterminate form in precalculus. Direct substitution gives 0/0, which is meaningless — but the limit may still exist. Fix with algebraic simplification: factor, cancel, rationalize.
∞/∞ form: Appears with rational functions as x→±∞. Divide numerator and denominator by the highest power of x.
∞ − ∞ form: Combining limits that both blow up. Requires algebraic manipulation to resolve.
0 · ∞ form: Rewrite as a quotient to convert to 0/0 or ∞/∞.
One-on-one Precalculus tutoring builds genuine intuition for why the limit laws work, how to handle indeterminate forms, and how limits connect to continuity and derivatives. We work through your actual homework and exam problems.