AcademyFoundations And Limits
Academy
Limit Laws
Level 1 - Calculus topic page in Foundations And Limits.
Principle
A limit describes the value a function approaches as the input approaches a point. Limit laws let you combine simpler limits to evaluate more complicated expressions without rebuilding the definition every time.
Notation
\(\lim_{x\to a} f(x)\)
limit of f(x) as x approaches a
\(a\)
approach value of the input
\(L,M\)
finite limit values
The Core Method
If \(\lim_{x\to a}f(x)=L\) and \(\lim_{x\to a}g(x)=M\), then sums, differences, constant multiples, products, and quotients can be handled term-by-term when the quotient denominator limit is non-zero.
Product law
\[\lim_{x\to a}f(x)g(x)=LM\]
For polynomials and rational functions with non-zero denominator at \(a\), direct substitution is usually valid. If substitution gives \(0/0\), simplify first.
Worked Cases
Question
Find \(\lim_{x\to2}\frac{x^2-4}{x-2}\).
Answer
Direct substitution gives \(0/0\), so factor the numerator: \(x^2-4=(x-2)(x+2)\). For \(x\ne2\), the expression equals \(x+2\). Therefore the limit is \(2+2=4\).
Examples
Question
Find \(\lim_{x\to1}(3x^2-2x+5)\).
Answer
Polynomials are continuous, so substitute \(x=1\): \(3(1)^2-2(1)+5=3-2+5=6\).