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Concept module

Limits and Continuity / Approaching a Value

Approach one target point from the left and right, compare the limiting height with the actual function value, and contrast continuous, removable, jump, and blow-up behavior on one honest graph.

The simulation shows one coordinate graph with a dashed target line at x = 0, a function curve, and two movable sample points approaching the target from the left and the right. Depending on the selected case, the graph can stay continuous, show a removable hole with a separate filled point, split into a jump, or rise and fall sharply on opposite sides like a vertical asymptote. A side card reports the current left-hand value, right-hand value, limit read, and actual function value at the target. Optional guides mark the sample points, the finite limiting height when one exists, and the actual point or undefined status at x = 0. At distance h = 0.16, the left-hand value is 0.84 and the right-hand value is 0.96. Both sides are approaching the same finite height of 0.9, but the actual function value at the target is 2.4, so the graph has a removable discontinuity.

Interactive lab

Limits and continuity

Drag either sample point or use the case and distance controls.

-3-2-10123-6-4-20246xf(x)target x = 0Visual cuesActive graphFrom the leftFrom the rightFinite limitActual f(0)Approach readoutcaseRemovable holecontinuityremovable discontinuityh0.16left0.84right0.96limit0.9f(0)2.4Both sides are closing in on the same height.The point at x = 0 sits somewhere else, so continuity breaks.

Controls

Removable hole
0.16

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ComparePrompt 1 of 3
Switch between the continuous and removable-hole cases. The nearby values can approach the same height in both cases even though only one keeps the filled point in the same place.

Graphs

Switch graph views without breaking the live stage and time link.

One-sided approach

Shows how the left-hand and right-hand values behave as the sample points move inward toward the target.

distance to target: 0.08 to 1.8value: -6.4 to 6.4
From the leftFrom the rightFinite limitActual f(0)
One-sided approachShows how the left-hand and right-hand values behave as the sample points move inward toward the target.0.080.510.941.371.8-6.4-3.203.26.4distance to targetvalue
Hover or scrub to link the graph back to the stage.distance to target / value

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

Behavior case
1

Switches between a continuous graph, a removable hole, a jump, and an asymptote-like blow-up while keeping the same target x-value.

Graph: One-sided approachOverlay: Approach markersOverlay: Finite-limit guideOverlay: Actual point

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use the live prompts to keep one-sided approach, finite limit, and continuity on the same graph story.

ComparePrompt 1 of 3
Graph: One-sided approach
Switch between the continuous and removable-hole cases. The nearby values can approach the same height in both cases even though only one keeps the filled point in the same place.
Control: Behavior caseGraph: One-sided approachOverlay: Finite-limit guideOverlay: Actual pointEquation

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

3 visible

Overlay focus

Approach markers

Show the left-hand and right-hand sample points together with their guide lines.

What to notice

  • Moving the markers inward makes the one-sided behavior easier to compare honestly.

Why it matters

It keeps limits tied to visible nearby values rather than to a slogan about x reaching the point.

Control: Distance to targetGraph: One-sided approachEquation

Challenge mode

Use the one-sided approach graph as a classification bench. The goal is to catch the difference between agreeing on a limit and actually being continuous at the target.

0/1 solved
ConditionCoreSolved

6 of 6 checks

Continuity classification checkpoint

Switch to the case where both one-sided values nearly agree, but the graph still is not continuous at .
Graph-linkedGuided start2 hints

Suggested start

Start from the fully continuous case, then change only what is needed to make the limit survive while continuity fails.
Matched
Open the One-sided approach graph.
One-sided approach
Matched
Keep the Approach markers visible.
On
Matched
Keep the Finite-limit guide visible.
On
Matched
Keep the Actual point visible.
On
Matched
Select the removable-hole case by moving the behavior case to the band between and .
1
Matched
Shrink the sample distance into the close-reading band from to .
0.16

Challenge solved

You found the removable-hole classification cleanly. The one-sided values agree on one finite limit, but the actual point still breaks continuity for a visible reason.
Read next: Integral as Accumulation / Area
At distance h = 0.16, the left-hand value is 0.84 and the right-hand value is 0.96. Both sides are approaching the same finite height of 0.9, but the actual function value at the target is 2.4, so the graph has a removable discontinuity.
Equation detailsDeeper interpretation, notes, and worked variable context.

One-sided limits

Read what the graph approaches from the left and from the right of the target.

Behavior case 1 Approach distance 0.16

Two-sided limit

A finite two-sided limit exists only when both one-sided values settle toward the same number L.

Behavior case 1 Approach distance 0.16

Continuity at a point

A function is continuous at the target only when the limiting value matches the actual function value there.

Behavior case 1

Progress

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Let the live model runChange one real controlOpen What to notice

Try this setup

Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.

This bench opened from a setup link. Any new copy will reflect the state you see now.

Current bench

Removable hole preset

This bench still matches one named preset, so the copied link will reopen that same starting point along with the current graph, overlays, and inspect context.

Open default bench

Featured setups

Continuous match

Open the case where both sides and the actual point land on the same height.

Removable hole

Open the same finite limit with the actual point moved away from it.

Jump split

Open the case where the left-hand and right-hand sides settle toward different heights.

Blow-up

Open the asymptote-like case where the nearby values do not settle to any finite height.

Saved setups

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Stable links

Starter track

Step 5 of 60 / 6 complete

Functions and Change

Earlier steps still set up Limits and Continuity / Approaching a Value.

1. Graph Transformations2. Rational Functions / Asymptotes and Behavior3. Exponential Change / Growth, Decay, and Logarithms4. Derivative as Slope / Local Rate of Change+2 more steps

Previous step: Derivative as Slope / Local Rate of Change.

Start from track beginning

Short explanation

What the system is doing

A limit is easier to trust when it stays visual. This bench keeps one target x-value fixed, lets you approach it from the left and the right, and shows whether the graph is settling toward one shared height, two different heights, or an asymptote-like blow-up.

Continuity is the second visual check. Even if both sides head toward the same height, the graph is only continuous there when the actual filled point lands on that same value. A removable hole, a jump, and a blow-up break continuity for different visible reasons.

Key ideas

01One-sided limits ask what the graph is doing as x approaches the target from the left or from the right.
02A finite two-sided limit exists only when both one-sided values settle toward the same height.
03Continuity at a point means the limiting value and the actual function value agree at that point.

Worked example

Read the full frozen walkthrough.

Frozen walkthrough
Use the current case and approach distance from the live bench so one-sided approach, finite limits, and continuity stay tied to the graph you are actually inspecting.

Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.

View plans
Frozen valuesUsing frozen parameters

For the current Removable hole case with , what do the left-hand and right-hand values suggest about the limit story?

Behavior case

Removable hole

Approach distance

0.16

1. Read the active case and distance

The bench is showing the Removable hole case with sample points at from .

2. Read the two one-sided values

From the graph, the left-hand sample is about 0.84 and the right-hand sample is about 0.96.

3. Decide what that says about the limit story

The one-sided traces are settling toward the same finite height, because the left-hand limit is 0.9 and the right-hand limit is 0.9.

One-sided approach read

The nearby values are agreeing on one shared height, so the graph supports a finite two-sided limit even before you check whether the actual point matches it.

Common misconception

If a graph has a filled point at x = a, then the limit there must match that point automatically.

The limit is about what nearby x-values are approaching, not about the point being filled in.

A removable hole can have a perfectly clear finite limiting value while the actual function value sits somewhere else.

Mini challenge

Switch to a case where the left-hand and right-hand values nearly agree, but continuity still fails.

Prediction prompt

Decide whether you need a removable hole, a jump, or a blow-up before you test it.

Check your reasoning

You need the removable-hole case, where both sides can settle toward one shared height even though the actual point is placed somewhere else.
That is the cleanest visual split between having a finite limit and being continuous. The approach behavior can be well behaved even when the plotted point breaks the match.

Quick test

Graph reading

Question 1 of 3

Answer from the graph behavior, not from a memorized slogan.

What is the cleanest condition for a finite two-sided limit at x = a?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows one coordinate graph with a dashed target line at x = 0, a function curve, and two movable sample points approaching the target from the left and the right. Depending on the selected case, the graph can stay continuous, show a removable hole with a separate filled point, split into a jump, or rise and fall sharply on opposite sides like a vertical asymptote.

A side card reports the current left-hand value, right-hand value, limit read, and actual function value at the target. Optional guides mark the sample points, the finite limiting height when one exists, and the actual point or undefined status at x = 0.

Graph summary

The one-sided-approach graph plots value against distance to the target, so the left-hand and right-hand traces show whether the two sides are converging to the same finite number, separating into a jump, or blowing up without bound.

Optional horizontal guides keep the finite limit and the actual function value visible as separate objects, which makes removable holes and true continuity easy to compare.