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Learn through experiments

Nearby behavior is not the value at the point

Can a function approach a number where its own value is undefined?

Entering midway? A function rule has a domain. Dividing by zero is undefined; simplifying an expression must preserve excluded inputs.

1 · Try

Start with a concrete question

Compare a rational function with a missing point and a step function with a jump. Approach x=1 from both directions.

Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.

Go directly to the experiment · Review prerequisite mathematics and return here

2 · Predict

What do you expect?

Will a smaller interval make the two sides agree in both examples?

3 · Experiment

Change one thing. Inspect what happens.

  1. Inspect the rational rule at distances 1, 0.1 and 0.01.
  2. Explain why cancellation gives nearby values but does not define the original value at x=1.
  3. Switch to the step and compare both one-sided values as the distance shrinks.
  4. Write separate statements about the function value and two-sided limit.
Experiment inputs
Inputs
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Distance to x=10.01
Value approaching from left1.99
Value approaching from right2.01
Side-to-side gap0.02

Approaching a missing point or jump without connecting across it

Horizontal: Input x. Vertical: Function value.

-0.5-111.52.54Current side samples: (0.99, 1.99)Current side samples: (1.01, 2.01)
  • 1. Left branch (x<1) (solid line)
  • 2. Right branch (x>1) (dashed line)
  • 3. Current side samples (points)

Axes use the displayed ranges.

Read plotted values as a table
SeriesInput xFunction value
Left branch (x<1)-0.50.5
Left branch (x<1)-0.47020.5298
Left branch (x<1)-0.44040.5596
Left branch (x<1)-0.41060.5894
Left branch (x<1)-0.38080.6192
Left branch (x<1)-0.3510.649
Left branch (x<1)-0.32120.6788
Left branch (x<1)-0.29140.7086
Left branch (x<1)-0.26160.7384
Left branch (x<1)-0.23180.7682
Left branch (x<1)-0.2020.798
Left branch (x<1)-0.17220.8278
Left branch (x<1)-0.14240.8576
Left branch (x<1)-0.11260.8874
Left branch (x<1)-0.08280.9172
Left branch (x<1)-0.0530.947
Left branch (x<1)-0.02320.9768
Left branch (x<1)0.00661.007
Left branch (x<1)0.03641.036
Left branch (x<1)0.06621.066
Left branch (x<1)0.0961.096
Left branch (x<1)0.12581.126
Left branch (x<1)0.15561.156
Left branch (x<1)0.18541.185
Left branch (x<1)0.21521.215
Left branch (x<1)0.2451.245
Left branch (x<1)0.27481.275
Left branch (x<1)0.30461.305
Left branch (x<1)0.33441.334
Left branch (x<1)0.36421.364
Left branch (x<1)0.3941.394
Left branch (x<1)0.42381.424
Left branch (x<1)0.45361.454
Left branch (x<1)0.48341.483
Left branch (x<1)0.51321.513
Left branch (x<1)0.5431.543
Left branch (x<1)0.57281.573
Left branch (x<1)0.60261.603
Left branch (x<1)0.63241.632
Left branch (x<1)0.66221.662
Left branch (x<1)0.6921.692
Left branch (x<1)0.72181.722
Left branch (x<1)0.75161.752
Left branch (x<1)0.78141.781
Left branch (x<1)0.81121.811
Left branch (x<1)0.8411.841
Left branch (x<1)0.87081.871
Left branch (x<1)0.90061.901
Left branch (x<1)0.93041.93
Left branch (x<1)0.96021.96
Left branch (x<1)0.991.99
Right branch (x>1)1.012.01
Right branch (x>1)1.042.04
Right branch (x>1)1.072.07
Right branch (x>1)1.0992.099
Right branch (x>1)1.1292.129
Right branch (x>1)1.1592.159
Right branch (x>1)1.1892.189
Right branch (x>1)1.2192.219
Right branch (x>1)1.2482.248
Right branch (x>1)1.2782.278
Right branch (x>1)1.3082.308
Right branch (x>1)1.3382.338
Right branch (x>1)1.3682.368
Right branch (x>1)1.3972.397
Right branch (x>1)1.4272.427
Right branch (x>1)1.4572.457
Right branch (x>1)1.4872.487
Right branch (x>1)1.5172.517
Right branch (x>1)1.5462.546
Right branch (x>1)1.5762.576
Right branch (x>1)1.6062.606
Right branch (x>1)1.6362.636
Right branch (x>1)1.6662.666
Right branch (x>1)1.6952.695
Right branch (x>1)1.7252.725
Right branch (x>1)1.7552.755
Right branch (x>1)1.7852.785
Right branch (x>1)1.8152.815
Right branch (x>1)1.8442.844
Right branch (x>1)1.8742.874
Right branch (x>1)1.9042.904
Right branch (x>1)1.9342.934
Right branch (x>1)1.9642.964
Right branch (x>1)1.9932.993
Right branch (x>1)2.0233.023
Right branch (x>1)2.0533.053
Right branch (x>1)2.0833.083
Right branch (x>1)2.1133.113
Right branch (x>1)2.1423.142
Right branch (x>1)2.1723.172
Right branch (x>1)2.2023.202
Right branch (x>1)2.2323.232
Right branch (x>1)2.2623.262
Right branch (x>1)2.2913.291
Right branch (x>1)2.3213.321
Right branch (x>1)2.3513.351
Right branch (x>1)2.3813.381
Right branch (x>1)2.4113.411
Right branch (x>1)2.443.44
Right branch (x>1)2.473.47
Right branch (x>1)2.53.5
Current side samples0.991.99
Current side samples1.012.01

Follow the calculation

  1. Choose h=10^−2=0.01; evaluate at 1−h=0.99 and 1+h=1.01.
  2. For x≠1, (x²−1)/(x−1)=(x−1)(x+1)/(x−1)=x+1. At x=1, the original expression is 0/0 and remains undefined.
  3. Left value 1.99000; right value 2.01000.
  4. Both approach 2 although the function has no value at 1. Exact error on each side is h=0.01.

The limit is 2; a limit describes nearby behavior and need not equal an existing function value.

Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.

4 · Understand

Explain the mechanism

Start with the meaning

A function rule has a domain. Dividing by zero is undefined; simplifying an expression must preserve excluded inputs.

Compare a rational function with a missing point and a step function with a jump. Approach x=1 from both directions.

A worked explanation

A limit asks what outputs approach as inputs approach a point. It does not require evaluating the rule at that point.

For x≠1, factor x²−1=(x−1)(x+1). At 0.99 and 1.01 the outputs are 1.99 and 2.01, approaching 2. The original rule remains undefined at 1.

The step has left limit zero and right limit two. No single two-sided limit exists, regardless of how small the approach distance becomes.

Symbols and relationship

lim as x→1 of (x²−1)/(x−1)=2, while f(1) is undefined. A two-sided limit requires matching left and right limits.

Use the graph and trace together

The graph and calculations use your current inputs. Read axis labels and units before comparing positions or slopes. Change one input, predict the effect, and use the numerical trace to verify the plotted behavior. A displayed curve is an illustration; the formula and its assumptions determine which conclusions are justified.

Scope and edge cases

  • Finite samples illustrate a limit; the algebra and one-sided definitions establish the result.
  • The plot leaves a gap around the excluded point and does not connect across the jump.

Return to the course and cumulative project

5 · Apply

Try a fresh case

For x≠3, g(x)=(x²−9)/(x−3). What is its limit as x approaches 3?

This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.

Take the investigation further

Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.

School investigation

Construct a table approaching a hole from both sides.

College investigation

Prove the rational limit using a direct epsilon–delta bound after cancellation.

Workplace investigation

Explain why a removable formula singularity and a genuine discontinuity need different software handling.

6 · Continue

Keep the evidence, choose your next step

Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.