Unit 1 · Lesson 4 of 8

A limit and a value must agree.

A sensor calibration should not produce an isolated jump in output. Move the assigned value and see whether it joins the surrounding response.

Change the input. Follow the mathematics.

What you'll be able to do

Set up the model and calculate its absolute mismatch with the limit.
Connect the equation to the changing visual.
Explain the assumptions and identify a misleading interpretation.

Predict before exploring

What will change if you increase value assigned at x=2? Predict the direction, then explain the response.

A model you can changeMove one control at a time.

Read the model

f(x)=\begin{cases}x+2&x\ne2\\c&x=2\end{cases},\quad\text{continuous at 2 iff }c=4

The left and right limits are both 4. Only the single assigned value changes.

Nearby values

Value assigned at x=26

absolute mismatch with the limit

|f(2)-\lim_{x\to2}f(x)|=|6-4|=2

Continuity requires a defined value, an existing two-sided limit, and agreement between them.

Assumptions, units, and limits

The left and right limits are both 4. Only the single assigned value changes.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

From the situation to an equation

At a point, continuity requires a defined value, an existing two-sided limit, and equality between them. A removable discontinuity can be repaired by setting the missing or mismatched value equal to its limit.

f(x)=\begin{cases}x+2&x\ne2\\c&x=2\end{cases},\quad\text{continuous at 2 iff }c=4

Keep the reasoning visible

\lim_{x\to a}f(x)=f(a)

Continuity requires a defined value, an existing two-sided limit, and agreement between them.

What makes this model useful?

The left and right limits are both 4. Only the single assigned value changes.

Matching only one side is insufficient. For a piecewise function check both one-sided limits and the assigned value.

A worked example

Calculate the starting scenario

Use value assigned at x=2 = 6 .
$|f(2)-\lim_{x\to2}f(x)|=|6-4|=2$
Continuity requires a defined value, an existing two-sided limit, and agreement between them. The checked quantity is absolute mismatch with the limit.

Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the absolute mismatch with the limit using value assigned at $x=2$ 6 . State the meaning of its sign and units.

Checked quantity: absolute mismatch with the limit · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

f(x)=\begin{cases}x+2&x\ne2\\c&x=2\end{cases},\quad\text{continuous at 2 iff }c=4

The left and right limits are both 4. Only the single assigned value changes.

Nearby values

Value assigned at x=26

absolute mismatch with the limit

|f(2)-\lim_{x\to2}f(x)|=|6-4|=2

Continuity requires a defined value, an existing two-sided limit, and agreement between them.

Assumptions, units, and limits

The left and right limits are both 4. Only the single assigned value changes.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

f(x)=\begin{cases}x+2&x\ne2\\c&x=2\end{cases},\quad\text{continuous at 2 iff }c=4

Calculate and explain

Apply the idea to a new scenario

Set the assigned value at $x=2$ to 4. Find the mismatch with the limit.

Checked quantity: absolute mismatch with the limit · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

f(x)=\begin{cases}x+2&x\ne2\\c&x=2\end{cases},\quad\text{continuous at 2 iff }c=4

The left and right limits are both 4. Only the single assigned value changes.

Nearby values

Value assigned at x=24

absolute mismatch with the limit

|f(2)-\lim_{x\to2}f(x)|=|4-4|=0

Continuity requires a defined value, an existing two-sided limit, and agreement between them.

Assumptions, units, and limits

The left and right limits are both 4. Only the single assigned value changes.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

f(x)=\begin{cases}x+2&x\ne2\\c&x=2\end{cases},\quad\text{continuous at 2 iff }c=4

Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual

Read the model

f(x)=\begin{cases}x+2&x\ne2\\c&x=2\end{cases},\quad\text{continuous at 2 iff }c=4

The left and right limits are both 4. Only the single assigned value changes.

Nearby values

Value assigned at x=26

absolute mismatch with the limit

|f(2)-\lim_{x\to2}f(x)|=|6-4|=2

Continuity requires a defined value, an existing two-sided limit, and agreement between them.

Assumptions, units, and limits

The left and right limits are both 4. Only the single assigned value changes.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. At a point, continuity requires a defined value, an existing two-sided limit, and equality between them. A removable discontinuity can be repaired by setting the missing or mismatched value equal to its limit.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

Completion is saved on this browser. This is separate from account practice progress.