Laboratory · Limits
Limits: the approach laboratory
Move x and observe f(x): distinguish the point value from the limit, compare both sides and seek justification beyond the numbers.
Choose an example or enter a function.
The existence of a limit is not decided automatically. Samples may miss oscillations or features; an unavailable value may reflect the domain or numerical precision. An entered L is only a hypothesis.
FunctionLeft sampleRight / infinity sampleLimit or hypothesis L
Near zero the oscillations are closer than the pixels: the graph cannot show all of them.
Left-hand limit
Right-hand limit
Point value f(a)
Conclusion
Why it works
Numerically stable guided evaluation: an algebraically equivalent expression is used at allowed inputs.
Two sequences: the same input limit, different outputs
x_n and z_n approach zero from the right, but f(x_n)=1 and f(z_n)=-1 for every n.
Compare the sample table
Approximate numerical values; a finite table is not a proof.