The simplest approach to any limit is direct substitution: plug in the value and compute. For polynomials, this always works. For rational functions away from zeros of the denominator, it works just as well. But substitution has limits of its own.
When plugging in produces 0/0, the expression is indeterminate—neither the numerator nor denominator alone determines the result. The limit might be any finite number, or infinite, or nonexistent. The form 0/0 signals that cancellation is hiding the true behavior, and algebraic work is required to reveal it.
This page covers the core techniques: factoring, rationalizing, and algebraic manipulation. Each method transforms an indeterminate expression into one where substitution succeeds.
Direct Substitution — Try This First
Before attempting any technique, substitute a into f(x) directly. If the result is a finite number with no division by zero, that number is the limit:
x→alimf(x)=f(a)
This works for:
a
Direct substitution exploits continuity. When f is continuous at a, the limit equals the function value by definition.
When Direct Substitution Fails
Substitution fails when it produces an undefined or indeterminate expression. The most common outcome is 0/0: both numerator and denominator evaluate to zero.
For example:
x→2limx−2x2−4
Substituting x=2 gives 00. This does not mean the limit is zero, undefined, or nonexistent. It means the expression's behavior near x=2 is not yet determined—more work is needed.
The 0/0 form indicates that both numerator and denominator share a common factor of (x−2). Removing this factor reveals the limit.
Indeterminate Forms
Several forms signal that limit rules cannot be applied directly:
00∞∞0⋅∞∞−∞
001∞∞0
Each form represents a competition between opposing tendencies. In 0/0, both numerator and denominator vanish—which vanishes faster determines the limit. In ∞−∞, both terms grow without bound—their difference depends on relative growth rates.
Indeterminate forms require transformation. The goal is to rewrite the expression so that substitution or limit rules apply.
Factoring and Canceling
When substitution yields 0/0, the numerator and denominator share a common factor. Factor both, cancel the shared factor, then substitute.
x→2limx−2x2−4
Factor the numerator:
=x→2limx−2(x−2)(x+2)
Cancel (x−2):
=x→2lim(x+2)=4
The cancellation is valid because the limit considers x near 2, not at 2. For x=2, the factor (x−2) is nonzero and cancels legitimately.
Rationalizing — Conjugate Multiplication
When radicals appear and substitution fails, multiply by the conjugate to eliminate the radical.
x→0limxx+1−1
Substituting gives 0/0. Multiply numerator and denominator by x+1+1:
=x→0limx(x+1+1)(x+1−1)(x+1+1)
The numerator becomes a difference of squares:
=x→0limx(x+1+1)(x+1)−1=x→0limx(x+1+1)x
Cancel x:
=x→0limx+1+11=21
Expanding and Simplifying
Sometimes expanding a product or simplifying a complex fraction reveals the cancellation needed.
x→1limx−1(x+1)2−4
Expand the numerator:
=x→1limx−1x2+2x+1−4=x→1limx−1x2+2x−3
Factor:
=x→1limx−1(x−1)(x+3)=x→1lim(x+3)=4
The initial form obscured the factor of (x−1); expansion made it visible.
Combining Fractions
When the expression involves a difference of fractions, combine them over a common denominator.
x→1lim(x−11−x2−12)
Note that x2−1=(x−1)(x+1). Rewrite with common denominator:
=x→1lim((x−1)(x+1)x+1−(x−1)(x+1)2)
=x→1lim(x−1)(x+1)x+1−2=x→1lim(x−1)(x+1)x−1
Cancel (x−1):
=x→1limx+11=21
Multiplying by Strategic Forms of 1
For limits at infinity, divide numerator and denominator by the highest power of x in the denominator.
x→∞lim2x2−73x2+5x−1
Divide every term by x2:
=x→∞lim2−x273+x5−x21
As x→∞, terms with x in the denominator vanish:
=2−03+0−0=23
This technique isolates the dominant terms that control behavior at infinity.
Using Known Limits
Recognize when parts of an expression match special limits and rewrite accordingly.