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.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Simplify without losing a domain restriction.
- Compare both approaches to a point.
- Choose a value that restores continuity.
Before you start
Factor a difference of squares and evaluate linear expressions.
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
A fictional calibration rule is f(x) = (x² − 9)/(x − 3), defined only when x ≠ 3. What output does it approach near 3, and how could it be made continuous?
Why this math matters
A missing measurement and an unpredictable neighborhood are different problems. Limits study nearby behavior, even if the formula cannot be evaluated at the central input. This distinction is useful when simplifying algebra, investigating numerical instability, or deciding whether a missing point can be filled consistently.
Set up the model
A useful answer starts with clear assumptions:
- Inputs and outputs are dimensionless calibration numbers.
- The original rule excludes x = 3.
- A continuous extension is a new definition, not a claim about an actual sensor reading.
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.
Can a formula have a hole but still approach one value?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A fictional calibration rule is f(x) = (x² − 9)/(x − 3), defined only when x ≠ 3. What output does it approach near 3, and how could it be made continuous?
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.
Identify the excluded input
At x = 3: numerator = 0 and denominator = 0
The expression 0/0 is undefined. It signals that direct substitution does not answer the limit question; it is not itself the limit's value.
Simplify nearby values
(x² − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3) = x + 3, for x ≠ 3
Cancellation is legal away from the hole. For example, inputs 2.99 and 3.01 produce outputs 5.99 and 6.01.
Match the approaches and fill the hole
lim as x → 3 of f(x) = 6; define g(3) = 6
Both sides approach six. A new function equal to f away from three and equal to six at three is continuous there.
The result
The original function has no value at 3, but its limit is 6. Assigning 6 creates a continuous extension.
If someone assigned the value ten at the missing point, the surrounding limit would remain six. That new function would still be discontinuous because its point value would not match the limit.
Common mistakes to catch
- Cancellation does not erase the original domain exclusion.
- A table supports intuition but does not replace the algebraic argument establishing this 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
Find the limit of (x² − 16)/(x − 4) as x approaches 4.
Show a hint
Factor the numerator into (x − 4)(x + 4).
Reveal answer and explanation
8
For x ≠ 4 the expression is x + 4, which approaches eight from both sides.
Practice 2
A rule is 1 when x < 0 and 2 when x > 0. Can assigning one value at zero make it continuous?
Show a hint
Compare the left and right limits.
Reveal answer and explanation
No
The one-sided limits are one and two. A single point value cannot make these different approaches agree.
Take the idea with you
Before evaluating a suspicious quotient numerically, inspect its domain and simplify it symbolically if possible.
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.
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