Describes the value a function approaches as the input approaches a certain point

Focuses on nearby behavior, not necessarily the value at the point itself

Foundation for defining derivatives (instantaneous rate of change)

Foundation for defining integrals (accumulation/area)

Supports rigorous treatment of continuity, infinite processes, and approximation

A function can get arbitrarily close to a value without ever reaching it

The limit can exist even if the function is undefined at the point

The limit can differ from the function’s value at the point

Table of values: evaluate near the point from both sides

Graph: observe the y-values as x gets close to the target x

Numerical approximation: compute with smaller and smaller distances to the point

lim(x→a) f(x) = L means:

ε (epsilon)

δ (delta)

0 < |x − a| < δ

No matter how strict the output closeness (ε) is, you can force it by choosing x sufficiently close to a (within δ)

lim(x→a) f(x) = L

Left-hand limit: lim(x→a−) f(x)

Right-hand limit: lim(x→a+) f(x)

Two-sided limit exists iff both one-sided limits exist and are equal

lim(x→∞) f(x) = L

lim(x→−∞) f(x) = L

lim(x→a) f(x) = ∞

lim(x→a) f(x) = −∞

Function values approach a single finite number from both sides

Left-hand and right-hand limits match

Values can be made arbitrarily close to the same L

Jump discontinuity

Infinite discontinuity

Oscillation

Path dependence (in multivariable limits)

Behavior

Typical pattern

Behavior

Typical pattern

Behavior

Typical pattern

Behavior

Typical pattern

lim(x→a) [f(x) + g(x)] = lim f + lim g

lim(x→a) [c·f(x)] = c·lim f

lim(x→a) [f(x)g(x)] = (lim f)(lim g)

lim(x→a) [f(x)/g(x)] = (lim f)/(lim g), if lim g ≠ 0

lim(x→a) [f(x)]^n = (lim f)^n

lim(x→a) √f(x) = √(lim f) when the limit is nonnegative and expressions are defined

If g is continuous at L and lim(x→a) f(x)=L, then lim(x→a) g(f(x)) = g(L)

f is continuous at a if:

Continuous functions preserve limits at points in their domain

Many common functions (polynomials, exponentials, trig) are continuous where defined

Works when f is continuous at the point

Often applies to polynomials and rational functions with nonzero denominator at a

Factoring and canceling common factors (removable discontinuities)

Rationalizing (especially with square roots)

Combining fractions to simplify complex expressions

lim(x→0) (sin x)/x = 1

lim(x→0) (1 − cos x)/x = 0

Use to evaluate more complex trig limits via rewriting

If h(x) ≤ f(x) ≤ k(x) near a and lim h = lim k = L, then lim f = L

Useful for oscillatory expressions bounded by shrinking envelopes

For indeterminate forms like 0/0 or ∞/∞

lim f/g = lim f’/g’ (under required conditions)

Typically used after confirming indeterminate form

Compare leading terms for rational functions

Divide numerator and denominator by highest power of x

Use growth-rate hierarchy (e.g., exponentials dominate polynomials)

0/0, ∞/∞

0·∞

∞ − ∞

0^0, 1^∞, ∞^0

The form alone does not determine the limit’s value

Requires algebraic transformation or specialized rules

Only works when the function is continuous at the point and defined appropriately

The limit depends on values near a, not necessarily at a

Two-sided limits require matching left and right behavior

Need consistent unbounded growth as x approaches the point, not just large values at some points

Defined via a limit of difference quotients as the increment approaches 0

Captures instantaneous rate of change/slope

Defined via a limit of Riemann sums as partition size approaches 0

Captures accumulation/area under a curve

Infinite sums and Taylor expansions depend on limit processes

Error control often relies on making terms small via limits