Math · Calculus

Limits and Continuity / Approaching a Value

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Derivative as Slope / Local Rate of ChangeUse the same approaching-language to turn secant slopes into a local derivative.

Key takeaway

Read left-hand and right-hand approach as separate evidence before naming a two-sided limit.
Explain why a removable hole can keep the same finite limit while failing continuity.
Distinguish a jump from blow-up by asking whether the nearby values settle to one finite height.
Use the continuity test in order: finite limit first, matching actual value second.

Common misconception

A filled point at the target does not decide the limit by itself; nearby approach behavior decides the limit, and the point only enters the continuity check.

The limit is about what nearby x-values are doing as they approach the target, not about the value at one point by itself.

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Reference and support

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Equation snapshotOne-sided limits · Two-sided limitShow 3 equations

One-sided limits
$lim_{x \to a^{-}} f (x), lim_{x \to a^{+}} f (x)$
Describe what the graph approaches from the left and from the right of the target.
Two-sided limit
$lim_{x \to a} f (x) = L$
A finite two-sided limit exists only when both one-sided limits approach the same number $L$ .
Continuity at a point
$lim_{x \to a} f (x) = f (a)$
Continuity at the target means the limit exists and equals the actual function value there.

Why it behaves this way

Explanation

A limit asks what the function values do as x gets close to a target, not what happens exactly at the target. On this bench the target stays fixed at $x = 0$ , and the case slider plus the distance control $h$ let you watch whether the left-hand and right-hand values approach one shared height, split toward different heights, or fail to settle to any finite height at all.

Continuity is the second question, not the first. First decide whether both sides approach one finite value. Then compare that limiting value with the actual plotted point at $x = 0$ : if they match, the function is continuous there; if they do not, you may have a removable hole, a jump, or no finite limit.

Key ideas

01Always read the left-hand and right-hand behavior separately first. A two-sided limit exists only when both sides approach the same value.

02A removable hole and a continuous graph can have the same finite limit. The difference is whether the actual function value at the target matches that limit.

03A jump fails because the two sides approach different heights, while a blow-up fails because the values do not settle to any finite height.

Worked examples

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Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the case slider, the distance

h

, and the one-sided markers together. First read what the two sides are doing. Then compare that limit story with the actual point at

x = 0

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the current Continuous case with $h = 0.55$ , what do the left-hand and right-hand markers suggest about the limit?

C

Behavior case

Continuous

h

Approach distance

0.55

1. Read the case and the sample distance

The bench is set to the Continuous case with sample markers at distance

h = 0.55

from

x = 0

2. Read the left-hand and right-hand values

From the graph, the left-hand sample is about 0.85 and the right-hand sample is about 1.35.

3. Decide whether a two-sided limit is possible

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 reading

f (0^{-}) \approx 0.85, f (0^{+}) \approx 1.35

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.

Quick test

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Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows one coordinate graph with a target 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 be continuous, show a removable hole with a separate filled point, split into a jump, or rise and fall sharply near the target like a vertical asymptote.

A side readout reports the current left-hand value, right-hand value, limit reading, 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 failing to settle at all.

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

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Matching limit and point

Open the case where both sides approach the same height and the actual point lands there too.

Same limit, different point

Open the case where both sides approach one height but the actual point is somewhere else.

Sides disagree

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

No finite landing height

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

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Starter track

Step 5 of 6

Functions and Change

Limits and Continuity / Approaching a Value appears later in this track, so it is cleaner to start from the beginning first.

Previous step: Derivative as Slope / Local Rate of Change

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