Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Use two converging bounds when a factor oscillates without a limit.
- Justify the conclusion "The limit is zero by the squeeze theorem" using the stated assumptions.
Before you start
Inequalities and basic limits.
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
Find lim(x→0)x²sin(1/x).
Why this math matters
Use two converging bounds when a factor oscillates without a limit. 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:
- The expression is evaluated for nonzero x.
- The sine function is bounded by one in magnitude.
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.
Control an oscillation by bounding its amplitude
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find lim(x→0)x²sin(1/x).
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
−1≤sin(1/x)≤1 for x≠0
The oscillation remains bounded despite becoming rapid.
Work through the mathematics
−x²≤x²sin(1/x)≤x²
Multiplication by the nonnegative x² preserves the inequality.
Check the conclusion
The limit is zero by the squeeze theorem
Both bounding functions approach zero, forcing the middle expression to do the same.
The result
The limit is zero by the squeeze theorem
Both bounding functions approach zero, forcing the middle expression to do the same.
Common mistakes to catch
- A factor without a limit can still appear in a product with a limit.
- Boundedness alone does not force a function to approach zero.
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
Does sin(1/x) alone have a limit at zero?
Show a hint
Choose inputs making its sine equal one and minus one.
Reveal answer and explanation
No
Persistent full-amplitude oscillations give incompatible subsequential values.
Practice 2
What about |x|sin(1/x)?
Show a hint
Bound it by ±|x|.
Reveal answer and explanation
Its limit is also zero
The shrinking absolute amplitude is sufficient.
Take the idea with you
Estimate an uncertain oscillating error using a reliable amplitude envelope.
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: Derive a power derivative from the difference quotient
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