Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Choose a delta that forces a requested output tolerance.
- Justify the conclusion "0<|x−2|<δ implies |(3x−1)−5|<ε" using the stated assumptions.
Before you start
Absolute values and the definition of a limit.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Prove lim(x→2)(3x−1)=5 using an epsilon–delta argument.
Why this math matters
Choose a delta that forces a requested output tolerance. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- x is real.
- ε is positive and arbitrary.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Turn a limit claim into an explicit error guarantee
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Prove lim(x→2)(3x−1)=5 using an epsilon–delta argument.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
|(3x−1)−5|=3|x−2|
Express output error directly in terms of input error.
Work through the mathematics
Given ε>0, choose δ=ε/3
The choice is made after the tolerance but before the input is selected.
Check the conclusion
0<|x−2|<δ implies |(3x−1)−5|<ε
This implication proves the limit for every positive requested tolerance.
The result
0<|x−2|<δ implies |(3x−1)−5|<ε
This implication proves the limit for every positive requested tolerance.
Common mistakes to catch
- δ may depend on ε but not on the later chosen x.
- Numerical sampling is not a proof for every nearby input.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Which δ works for ε=0.03?
Show a hint
Divide the tolerance by three.
Reveal answer and explanation
0.01
Multiplying an input error below 0.01 by three stays below 0.03.
Practice 2
Must δ be the largest possible choice?
Show a hint
The definition asks for existence.
Reveal answer and explanation
No
Any smaller positive delta also gives the implication.
Take the idea with you
Convert a linear measurement tolerance into a sufficient input precision.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Check convergence through distances between late terms
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