The limit of a function f as x approaches c is the value that f of x gets arbitrarily close to as x gets sufficiently close to c. The limit is about the approach, not about the actual value of the function at c. The function value at c can be anything, undefined, or even different from the limit, without changing what the limit is.

What is the difference between a one-sided and a two-sided limit?

A one-sided limit approaches c from only one direction, either from below or from above, written L minus or L plus. The two-sided limit requires that both one-sided limits exist, are finite, and are equal. When the one-sided limits disagree or one of them fails to exist, the two-sided limit does not exist.

When does a limit not exist?

A two-sided limit fails to exist in three main ways: when the left and right limits are finite but different (a jump), when at least one of them is plus or minus infinity (an infinite limit, typically a vertical asymptote), and when the function oscillates infinitely fast near the point so that no value is approached, as with sine of one over x near zero.

What is a removable discontinuity?

A removable discontinuity is a point where the two-sided limit exists cleanly, but the function value at that point is either undefined or set to something different from the limit. The discontinuity can be removed by simply redefining the function value at that single point to equal the limit, making the function continuous there.

How does this tool help understand limits?

The tool pairs the geometric picture of the function curve with the analytic values of the left and right limits. Sliding epsilon shows numerically that f of x approaches the limit values as x approaches c, which is exactly the formal definition. The seven function families cover every canonical limit behavior in a first calculus course, from continuous to oscillating.

Limit Explorer - Interactive Visualizer

Holef(x) = (x² − 1) / (x − 1)

880 × 460

f(x)L⁻ and L⁺x = capproach

From the leftx → c⁻

x =0.5

f(x) =1.5

↓ as ε → 0

L⁻ =2

From the rightx → c⁺

x =1.5

f(x) =2.5

↓ as ε → 0

L⁺ =2

Limit at x = cc = 1

Left limit L⁻

2

lim x→c⁻ f(x)

Right limit L⁺

2

lim x→c⁺ f(x)

Two-sided limit

2

L⁻ = L⁺

Limit exists. L⁻ and L⁺ agree at 2, so lim x→1 f(x) = 2.limit exists

f(c) = undefined. The limit exists but f isn’t defined at c. Set f(c) = 2 and the function becomes continuous.removable

Appliedc=1ε=0.5L⁻ = L⁺ ⇒ limit exists

Key Terms

Getting Started

The Discontinuity Zoo

The Epsilon Slider

The Approach Bar

The Three Limit Cards

The Two Verdict Rows

Display Toggles

What Is a Limit

One-Sided vs Two-Sided

Related Concepts

Key Terms

Limit at a point — the value

f(x)

approaches as

x

approaches

c

, written

\lim_{x \to c} f(x)

. The limit is about the *approach*, not the value

f(c)

itself.

Left limit —

L^{-} = \lim_{x \to c^{-}} f(x)

. The value approached from values smaller than

c

.

Right limit —

L^{+} = \lim_{x \to c^{+}} f(x)

. The value approached from values larger than

c

.

Two-sided limit — exists exactly when

L^{-}

and

L^{+}

are both finite and equal. The common value is the limit.

DNE — abbreviation for *does not exist*. Used for limits when the one-sided limits disagree, one is infinite, or the function oscillates.

Removable discontinuity — the two-sided limit exists but

f(c)

is either undefined or different from the limit. Patching

f(c)

would restore continuity.

Vertical asymptote — a value of

c

where at least one one-sided limit is

+\infty

-\infty

Getting Started

The page opens with the Hole family

f(x) = (x^2 - 1)/(x - 1)

already loaded, centered at

c = 1

. You see:

• The blue curve of

f

, with a gap at the hole.

• A dashed light-blue horizontal line at

L^{-} = L^{+} = 2

.

• A dashed gray vertical line at

x = c

.

• Two dots at

(c - \varepsilon, f(c - \varepsilon))

and

(c + \varepsilon, f(c + \varepsilon))

that move when you drag the

\varepsilon

slider.

Below the graph, the From the left and From the right cards report the numerical values approaching

L^{-}

and

L^{+}

\varepsilon

shrinks. The card below that gives the two-sided limit verdict and the continuity classification.

To explore quickly, switch families in the left panel — the discontinuity zoo covers every canonical case you'd meet in a first calculus course.

The Discontinuity Zoo

Seven function families are organized into five groups:

Continuous (control):

• Quadratic —

f(x) = x^2

c = 1

. Everything agrees; the baseline.

Removable:

• Hole —

(x^2 - 1)/(x - 1)

c = 1

. Limit exists;

f(c)

is undefined.

Jump:

• Step — piecewise

x

on the left and

x + 1

on the right of

c = 0

. Finite one-sided limits, different values.

Infinite:

• 1/x² — both one-sided limits go to

+\infty

c = 0

.

• 1/x — left limit goes to

-\infty

, right to

+\infty

c = 0

.

Oscillating:

• sin(1/x) — neither one-sided limit exists at

c = 0

. The function bounces infinitely fast.

One-sided:

• Square root —

\sqrt{x}

c = 0

. Only defined on the right; the left limit is undefined.

The Epsilon Slider

The distance ε slider controls how close to

c

we probe. As you drag

\varepsilon

toward zero:

• The two approach dots move toward the vertical line at

c

.

• The numbers in the From the left and From the right cards march toward

L^{-}

and

L^{+}

.

• The marker positions on the graph close in on the limit lines.

The slider is on a logarithmic scale, ranging from

\varepsilon = 1

down to

\varepsilon = 10^{-3}

. Each tick is roughly an order of magnitude, so very small values of

\varepsilon

are easy to reach precisely.

This mirrors the formal definition of a limit: for the limit to equal

L

, the value

f(x)

must get *arbitrarily close* to

L

when

x

gets sufficiently close to

c

. Shrinking

\varepsilon

is the visual equivalent of "as close as you like".

The Reset button next to Parameters restores the default

\varepsilon = 0.5

The Approach Bar

Two cards below the graph track the numerical approach side by side.

From the left (

x \to c^{-}

) shows:

•

x = c - \varepsilon

, the current probing position on the left.

•

f(x)

at that position.

•

L^{-}

, the left limit value (the target).

From the right (

x \to c^{+}

) shows the same triple for the right side.

The middle "

\downarrow

\varepsilon \to 0

" arrow is a reminder that the

f(x)

row is the dynamic quantity converging to the

L

row below it. Watch the digits stabilize as

\varepsilon

shrinks.

This is the calculator-style version of the geometric picture above. When the limit exists, both

f(x)

rows converge to the same number. When one of them fails to settle (oscillating family) or runs off to infinity, the limit does not exist.

The Three Limit Cards

The boxed card below the approach bar displays the analytic limit values for the current family:

• Left limit L⁻ — the analytic value of

\lim_{x \to c^{-}} f(x)

. May be a finite number,

+\infty

-\infty

, or DNE.

• Right limit L⁺ — same for

\lim_{x \to c^{+}} f(x)

.

• Two-sided limit — the common value when

L^{-}

and

L^{+}

agree and are finite, otherwise DNE.

Reading the row tells you exactly which kind of behavior is happening at

c

:

• Both finite, equal → limit exists.

• Both finite, different → jump discontinuity.

• Either infinite → infinite discontinuity.

• Either DNE (analytical, not numerical) → oscillation or undefined-on-a-side.

These are the analytic values the approach bar should be converging to as

\varepsilon

shrinks.

The Two Verdict Rows

Below the three cards are two verdict rows that compress the analysis into plain language.

Limit verdict — names the limit's status with a tag:

• limit exists — both one-sided limits agree on a finite value.

• DNE (jump) — finite but different one-sided limits.

• DNE (infinite) — at least one limit is

\pm\infty

.

• DNE (oscillating) — neither one-sided limit settles, like

\sin(1/x)

near

0

.

• DNE (one-sided only) — function only defined on one side of

c

.

Continuity verdict — names the discontinuity at

c

:

• continuous — limit exists,

f(c)

defined, and they agree.

• removable — limit exists but

f(c)

doesn't match (or is undefined). Patching

f(c)

fixes it.

• jump, infinite, essential — the unpatchable cases. No single value of

f(c)

can repair the discontinuity.

Display Toggles

The Display section in the left panel lets you hide layers when one gets in the way:

• f(x) — toggles the function curve. Off, just the limit lines and approach markers remain.

• L⁻, L⁺ — toggles the horizontal limit reference lines and the hollow circles at

(c, L^{\pm})

. Off, the picture loses its analytic target.

• x = c — toggles the dashed vertical line at

c

. Off, the location of

c

is implied only by the markers and the family info.

• approach — toggles the two moving probe dots at

(c \pm \varepsilon, f(c \pm \varepsilon))

. Off, the picture is static; the slider still controls the approach card numbers.

The Accent color picker at the bottom recolors the active highlight throughout the tool — useful for screenshots or personal preference.

What Is a Limit

The limit of a function

f

x

approaches

c

is the single value (if any) that

f(x)

gets arbitrarily close to as

x

gets sufficiently close to

c

. Written:

\lim_{x \to c} f(x) = L

The crucial point is that the limit is about the approach, not the destination. The actual value

f(c)

— if it exists — is irrelevant to the limit. The limit asks only what value

f(x)

is heading toward as

x

closes in on

c

.

Formally (the

\varepsilon

–

\delta

definition): for every

\varepsilon > 0

there exists a

\delta > 0

such that

|f(x) - L| < \varepsilon

whenever

0 < |x - c| < \delta

. This formalizes "arbitrarily close" and "sufficiently close" with two tolerances chained together.

For the full theoretical treatment, see the limits page.

One-Sided vs Two-Sided

Approaching

c

from values smaller than

c

gives the left limit

L^{-} = \lim_{x \to c^{-}} f(x)

. From values larger than

c

gives the right limit

L^{+} = \lim_{x \to c^{+}} f(x)

.

The two-sided limit exists if and only if both one-sided limits exist, are finite, and are equal:

\lim_{x \to c} f(x) = L \iff L^{-} = L^{+} = L

So the two-sided limit fails to exist in three ways:

•

L^{-}

and

L^{+}

are both finite but unequal — a jump.

• At least one of

L^{-}

L^{+}

is infinite — an infinite limit.

• At least one of

L^{-}

L^{+}

fails to exist at all — an oscillating or one-sided case.

The tool's discontinuity zoo demonstrates each of these failure modes alongside the success case.

For deeper coverage of one-sided limits, see the one-sided limits page.

Related Concepts

Continuity — a function is continuous at

c

exactly when the limit at

c

exists,

f(c)

is defined, and they agree. The limit is one of the three conditions in the continuity definition. See the continuity checker.

Limit laws — the algebraic rules for combining limits: limits of sums, products, quotients, compositions. Most limit computations reduce to applying the laws and evaluating at the point.

Limits at infinity — the behavior of

f(x)

x \to \pm\infty

. Determines horizontal asymptotes.

Indeterminate forms — limits where direct substitution gives

0/0

\infty/\infty

0 \cdot \infty

, or

\infty - \infty

. Resolved by algebraic manipulation or L'Hôpital's rule.

Squeeze theorem — if

g(x) \le f(x) \le h(x)

near

c

and

\lim g = \lim h = L

, then

\lim f = L

. Useful for oscillating functions trapped between bounds.

Derivatives — defined as a limit of secant slopes. See the derivative visualizer.

Definite integrals — defined as limits of Riemann sums. See the Riemann sum visualizer.