Learn with Amar
Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Separate a function's assigned value from its nearby behavior.
- Justify the conclusion "The two-sided limit does not exist, regardless of f(0)=7" using the stated assumptions.
Before you start
Piecewise functions and 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
Let f(x)=1 for x<0, f(0)=7, and f(x)=3 for x>0. Does lim(x→0)f(x) exist?
Why this math matters
Separate a function's assigned value from its nearby behavior. 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 domain contains inputs on both sides of zero.
- The function follows exactly the stated piecewise rule.
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.
Diagnose a jump through unequal one-sided limits
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Let f(x)=1 for x<0, f(0)=7, and f(x)=3 for x>0. Does lim(x→0)f(x) exist?
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
The left-hand limit is one
Every negative input sufficiently near zero has output one.
Work through the mathematics
The right-hand limit is three
Positive nearby inputs use the other constant branch.
Check the conclusion
The two-sided limit does not exist, regardless of f(0)=7
A limit depends on nearby values agreeing, not on the isolated assigned value.
The result
The two-sided limit does not exist, regardless of f(0)=7
A limit depends on nearby values agreeing, not on the isolated assigned value.
Common mistakes to catch
- A function value and a limit are different objects.
- Averages of unequal one-sided limits do not define the two-sided limit.
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
Would changing f(0) to two fix the limit?
Show a hint
One-sided behavior is unchanged.
Reveal answer and explanation
No
The jump between one and three remains.
Practice 2
If both outside branches equaled three, what would the limit be?
Show a hint
Compare the two one-sided limits again.
Reveal answer and explanation
Three
The isolated value at zero could still differ without altering the limit.
Take the idea with you
Distinguish an isolated bad reading from a genuine jump in a response rule.
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: Control an oscillation by bounding its amplitude
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