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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.

Interactive lab

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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.

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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

Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

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.

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Frozen valuesUsing frozen parameters

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

Behavior case

Continuous

Approach distance

0.6

1. Read the active case and distance

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

2. Read the two one-sided values

From the graph, the left-hand sample is about 0.83 and the right-hand sample is about 1.37.

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 1.1 and the right-hand limit is 1.1.

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.

Make a prediction before you reveal the next step.

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

Check your reasoning against the live bench.

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?

Use the live bench to test the result before moving on.

Accessibility

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.