riaology
Unit 1 · Lesson 3 of 8

Follow a value toward a missing point.

A measurement formula can have a hole at one input while its nearby predictions approach a perfectly finite value.

Change the input. Follow the mathematics.

What you'll be able to do
  • Set up the model and calculate its nearby function value (or limit at h=0).
  • Connect the equation to the changing visual.
  • Explain the assumptions and identify a misleading interpretation.
Predict before exploring

What will change if you increase distance h from 2? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Read the modelf(x)=x2−4x−2=x+2 (x≠2),lim⁡x→2f(x)=4f(x)=\frac{x^2-4}{x-2}=x+2\ (x\ne2),\quad\lim_{x\to2}f(x)=4

The original function is undefined at x=2.x=2. At h=0h=0 the readout shows its limit, not a newly defined function value.

Input x versus Function value. Extension with a hole at 2. Selected nearby function value (or limit at h=0): 4.1 . Approach from either side. Nearby values can settle even when the value at the target is missing.003264Function valueInput x
Extension with a hole at 2

nearby function value (or limit at h=0)f(2+h)=4+h=4.1,lim⁡h→0f(2+h)=4f(2+h)=4+h=4.1,\qquad\lim_{h\to0}f(2+h)=4

Approach from either side. Nearby values can settle even when the value at the target is missing.

Assumptions, units, and limits

The original function is undefined at x=2.x=2. At h=0h=0 the readout shows its limit, not a newly defined function value.

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

01

From the situation to an equation

Factor the numerator and cancel only for inputs where the canceled factor is nonzero. The simplified expression describes nearby values, so it can establish a limit without defining the original function at the missing point.

f(x)=x2−4x−2=x+2 (x≠2),lim⁡x→2f(x)=4f(x)=\frac{x^2-4}{x-2}=x+2\ (x\ne2),\quad\lim_{x\to2}f(x)=4
02

Keep the reasoning visible

x2−4=(x−2)(x+2)x^2-4=(x-2)(x+2)

Approach from either side. Nearby values can settle even when the value at the target is missing.

03

What makes this model useful?

The original function is undefined at x=2.x=2. At h=0h=0 the readout shows its limit, not a newly defined function value.

Canceling a factor does not retroactively define the quotient where its denominator was zero.

A worked example

Calculate the starting scenario

  1. Use distance h from 2 = 0.1 .
  2. f(2+h)=4+h=4.1,lim⁡h→0f(2+h)=4f(2+h)=4+h=4.1,\qquad\lim_{h\to0}f(2+h)=4
  3. Approach from either side. Nearby values can settle even when the value at the target is missing. The checked quantity is nearby function value (or limit at h=0).
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 nearby function value (or limit at h=0h=0) using distance h from 2 0.1 . State the meaning of its sign and units.

Checked quantity: nearby function value (or limit at h=0) · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Factor the numerator and cancel only for inputs where the canceled factor is nonzero. The simplified expression describes nearby values, so it can establish a limit without defining the original function at the missing point.

Explore the visual solution
Read the modelf(x)=x2−4x−2=x+2 (x≠2),lim⁡x→2f(x)=4f(x)=\frac{x^2-4}{x-2}=x+2\ (x\ne2),\quad\lim_{x\to2}f(x)=4

The original function is undefined at x=2.x=2. At h=0h=0 the readout shows its limit, not a newly defined function value.

Input x versus Function value. Extension with a hole at 2. Selected nearby function value (or limit at h=0): 4.1 . Approach from either side. Nearby values can settle even when the value at the target is missing.003264Function valueInput x
Extension with a hole at 2

nearby function value (or limit at h=0)f(2+h)=4+h=4.1,lim⁡h→0f(2+h)=4f(2+h)=4+h=4.1,\qquad\lim_{h\to0}f(2+h)=4

Approach from either side. Nearby values can settle even when the value at the target is missing.

Assumptions, units, and limits

The original function is undefined at x=2.x=2. At h=0h=0 the readout shows its limit, not a newly defined function value.

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.

Factor the numerator and cancel only for inputs where the canceled factor is nonzero. The simplified expression describes nearby values, so it can establish a limit without defining the original function at the missing point.

f(x)=x2−4x−2=x+2 (x≠2),lim⁡x→2f(x)=4f(x)=\frac{x^2-4}{x-2}=x+2\ (x\ne2),\quad\lim_{x\to2}f(x)=4
Calculate and explain

Apply the idea to a new scenario

Use h=−0.2h=-0.2 and find the nearby value at x=2+hx=2+h. Compare with the two-sided limit.

Checked quantity: nearby function value (or limit at h=0) · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Factor the numerator and cancel only for inputs where the canceled factor is nonzero. The simplified expression describes nearby values, so it can establish a limit without defining the original function at the missing point.

Explore the visual solution
Read the modelf(x)=x2−4x−2=x+2 (x≠2),lim⁡x→2f(x)=4f(x)=\frac{x^2-4}{x-2}=x+2\ (x\ne2),\quad\lim_{x\to2}f(x)=4

The original function is undefined at x=2.x=2. At h=0h=0 the readout shows its limit, not a newly defined function value.

Input x versus Function value. Extension with a hole at 2. Selected nearby function value (or limit at h=0): 3.8 . Approach from either side. Nearby values can settle even when the value at the target is missing.003264Function valueInput x
Extension with a hole at 2

nearby function value (or limit at h=0)f(2+h)=4+h=3.8,lim⁡h→0f(2+h)=4f(2+h)=4+h=3.8,\qquad\lim_{h\to0}f(2+h)=4

Approach from either side. Nearby values can settle even when the value at the target is missing.

Assumptions, units, and limits

The original function is undefined at x=2.x=2. At h=0h=0 the readout shows its limit, not a newly defined function value.

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.

Factor the numerator and cancel only for inputs where the canceled factor is nonzero. The simplified expression describes nearby values, so it can establish a limit without defining the original function at the missing point.

f(x)=x2−4x−2=x+2 (x≠2),lim⁡x→2f(x)=4f(x)=\frac{x^2-4}{x-2}=x+2\ (x\ne2),\quad\lim_{x\to2}f(x)=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 modelf(x)=x2−4x−2=x+2 (x≠2),lim⁡x→2f(x)=4f(x)=\frac{x^2-4}{x-2}=x+2\ (x\ne2),\quad\lim_{x\to2}f(x)=4

The original function is undefined at x=2.x=2. At h=0h=0 the readout shows its limit, not a newly defined function value.

Input x versus Function value. Extension with a hole at 2. Selected nearby function value (or limit at h=0): 4.1 . Approach from either side. Nearby values can settle even when the value at the target is missing.003264Function valueInput x
Extension with a hole at 2

nearby function value (or limit at h=0)f(2+h)=4+h=4.1,lim⁡h→0f(2+h)=4f(2+h)=4+h=4.1,\qquad\lim_{h\to0}f(2+h)=4

Approach from either side. Nearby values can settle even when the value at the target is missing.

Assumptions, units, and limits

The original function is undefined at x=2.x=2. At h=0h=0 the readout shows its limit, not a newly defined function value.

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

The reasoning. Factor the numerator and cancel only for inputs where the canceled factor is nonzero. The simplified expression describes nearby values, so it can establish a limit without defining the original function at the missing point.

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.

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