Computing limits directly from definitions—chasing epsilons and deltas—is rigorous but impractical for everyday calculation. The power of limit rules lies in decomposition: break a complicated expression into simpler parts, find the limit of each part, then reassemble.
These rules reflect the algebraic structure of limits. The limit of a sum is the sum of the limits. The limit of a product is the product of the limits. Such properties hold because limits preserve algebraic operations, provided the component limits exist.
That final condition is crucial. Every rule listed here requires the individual limits to exist before the rule applies. When a limit yields an indeterminate form, the rules do not directly help—algebraic manipulation must come first.
Why Limit Rules Matter
Without rules, evaluating even simple limits would require returning to the definition each time. The definition asks whether f(x) can be made arbitrarily close to L by taking x sufficiently close to a. Proving this directly for each new function is tedious.
Limit rules provide shortcuts. Once you establish that basic limits exist—constants, the identity function, standard functions—you can combine them using algebraic rules to handle complex expressions.
The rules also reveal structure. They show that limits behave like algebraic operations in many ways, which is why calculus meshes so naturally with algebra.
Limit of a Constant
For any constant c:
x→alimc=c
A constant function f(x)=c outputs the same value regardless of input. As x approaches a, the output remains c. The limit is simply c.
This rule seems trivial but serves as a foundation. Combined with other rules, it handles constant terms in any expression.
Limit of the Identity Function
For the identity function f(x)=x:
x→alimx=a
As x approaches a, the value of x approaches a. This tautology provides the base case for handling any polynomial: every polynomial is built from constants and powers of x, and this rule handles the linear term.
Sum and Difference Rules
If limx→af(x)=L and limx→ag(x)=M, then:
x→alim[f(x)+g(x)]=L+M
x→alim[f(x)−g(x)]=L−M
The limit of a sum is the sum of the limits. The limit of a difference is the difference of the limits.
Both component limits must exist. If either fails to exist, these rules do not apply. The indeterminate form ∞−∞ illustrates what can go wrong: two quantities both growing without bound may have a difference that converges, diverges, or oscillates.
Constant Multiple Rule
If limx→af(x)=L and c is a constant:
x→alim[c⋅f(x)]=c⋅L
Constants factor out of limits. This is a special case of the product rule where one factor is constant, but it appears frequently enough to state separately.
Scaling a function by a constant scales its limit by the same constant.
Product Rule
If limx→af(x)=L and limx→ag(x)=M, then:
x→alim[f(x)⋅g(x)]=L⋅M
The limit of a product is the product of the limits. This extends to any finite number of factors: if each factor has a limit, the product's limit equals the product of those limits.
The indeterminate form 0⋅∞ shows the rule's limitation. When one factor approaches 0 while the other grows without bound, the product's behavior depends on the relative rates—it might approach 0, ∞, or any finite value.
Quotient Rule
If limx→af(x)=L and limx→ag(x)=M with M=0, then:
x→alimg(x)f(x)=ML
The limit of a quotient is the quotient of the limits, provided the denominator's limit is nonzero.
When M=0, the rule fails. If L=0 and M=0, the limit is typically infinite (analyze the sign to determine +∞ or −∞). If both L=0 and M=0, you have the indeterminate form 0/0, requiring further techniques.
Power Rule
If limx→af(x)=L and n is a positive integer:
x→alim[f(x)]n=Ln
Repeated application of the product rule yields this result. The limit of a power is the power of the limit.
For rational exponents:
x→alim[f(x)]m/n=Lm/n
provided L>0 (or L≥0 when n is odd). Root functions require care with domain restrictions.
Root Rule
If limx→af(x)=L and the n-th root exists:
x→alimnf(x)=nL
For odd n, this holds for all real L. For even n, the rule requires L≥0 and f(x)≥0 near a.
This is the power rule with exponent 1/n. The limit of a root is the root of the limit, provided the root is defined.
Absolute Value Rule
If limx→af(x)=L, then:
\lim_{x \to a} |f(x)| = |L|}
The limit of an absolute value is the absolute value of the limit.
The converse does not hold: \lim_{x \to a} |f(x)}| may exist even when limx→af(x) does not. For example, |(-1)^n|} equals 1 for all n, but (−1)n has no limit as n→∞.
Limits of Polynomials
For any polynomial p(x)=anxn+an−1xn−1+⋯+a1x+a0:
x→alimp(x)=p(a)
Direct substitution always works for polynomials. This follows from applying the sum, constant multiple, and power rules to each term.
Polynomials are continuous everywhere, so the limit at any point equals the function value at that point.
Limits of Rational Functions
For a rational function r(x)=q(x)p(x) where p and q are polynomials:
x→alimr(x)=q(a)p(a)provided q(a)=0
When the denominator is nonzero at a, substitute directly. When q(a)=0, the quotient rule fails and other techniques are needed.
If q(a)=0 but p(a)=0, the limit is infinite. If both p(a)=0 and q(a)=0, factor and cancel the common root before substituting.
The limit passes through continuous functions. You first find the limit of the inner function, then apply the outer function to that limit.
Continuity of f at L is essential. Without it, the behavior of f near L might not match f(L), and the rule fails.
The Squeeze Theorem
Suppose g(x)≤f(x)≤h(x) for all x near a (except possibly at a itself). If:
x→alimg(x)=Landx→alimh(x)=L
then:
x→alimf(x)=L
The function f is trapped between two functions that converge to the same limit. It has nowhere to go but L.
This theorem proves the special limitlimx→0xsinx=1 through a geometric argument that bounds xsinx between cosx and 1 near zero.
When Rules Fail
Every rule requires component limits to exist. When they don't, the rule cannot be applied directly.
Indeterminate forms signal this breakdown:
00 — quotient rule fails
∞∞ — quotient rule fails
0⋅∞ — product rule fails
∞−∞ — difference rule fails
00, 1∞, ∞0 — power rule fails
These forms require transformation before rules apply. Factor and cancel, rationalize, rewrite in equivalent forms—the goal is to eliminate the indeterminate form so that limit rules can finish the computation.
The evaluating limits page covers techniques for resolving these cases.