MATH1851 Chap.1 Limits and One-Sided Behaviour
Limits and One-Sided Behaviour
Approach is not value
A limit asks what outputs do when inputs move close to a target while remaining different from it. That is why a missing point, or a point plotted at an unexpected height, can leave the limit unchanged. The chapter begins with average rate of change and a secant line, then shrinks the input gap until the nearby slopes describe a tangent.
This geometric move gives the limit notation a purpose before algebra begins.
Existence is a two-direction agreement
Read the left-hand and right-hand limits separately. A finite two-sided limit exists only when both sides reach the same real number.
Unequal sides create a jump-type failure, endless oscillation prevents settling, and unbounded behaviour is recorded with an infinite-limit statement rather than a finite value. Classifying the failure tells you whether algebra can repair the expression or whether the graph has already decided the result.
Evaluation is a route choice
Start with direct substitution.
A finite result finishes the calculation when the function is continuous at the target. Zero over zero is not an answer; it signals factorisation, rationalisation or another equivalence on a punctured neighbourhood. A nonzero numerator over a vanishing denominator demands a one-sided sign check.
Numerical tables and graphs may suggest the destination, but a law or transformation supplies the justification.
Write the logical conclusion
End every calculation by stating the target, the direction of approach and the resulting value or failure type.
This short sentence distinguishes an algebraic simplification from the limit claim it justifies and keeps a one-sided result from being mistaken for a two-sided one.
What this chapter covers
- 01
Average and instantaneous rates of change
- 02
Nearby behaviour versus the function value
- 03
Left-hand and right-hand limits
- 04
Three ways a finite limit can fail
- 05
Limit laws and direct substitution
- 06
Factorisation and conjugate rationalisation
Worked example · free
Remove a hole without changing the limit
- 1Diagnose the indeterminate substitution result.
- 3Factor and simplify on the punctured neighbourhood.
- 2Use direct substitution on the simplified expression and state the limit.
Key terms
- Average Rate
- Output change divided by input change across an interval, represented by a secant slope.
- Instantaneous Rate
- The limiting rate at one input, represented by the tangent slope when the limit exists.
- Left-Hand Limit
- The limiting output obtained using inputs smaller than the target point.
- Right-Hand Limit
- The limiting output obtained using inputs larger than the target point.
- Indeterminate Form
- A substitution pattern such as zero over zero that signals more analysis rather than supplying the limit.
- Removable Hole
- A missing or mismatched point where a finite shared limit exists and one redefinition can restore continuity.
Limits and One-Sided Behaviour FAQ
Can a limit exist when the function is undefined at the point?
Yes. The limit uses nearby inputs and excludes the target itself. If the left-hand and right-hand outputs approach the same finite value, the limit exists even when no point value is defined. Assigning that limiting value would create a continuous extension.
What does zero over zero tell me?
It tells you direct substitution is inconclusive, not that the limit is zero or nonexistent. Look for a common factor, a conjugate rationalisation or another legal transformation that produces an equivalent expression for nearby nonzero inputs, then evaluate that expression.
When should I use a table or graph?
Use it to form a conjecture, identify one-sided behaviour and catch sign errors. Do not treat a finite sample or plotting window as proof. Complete the answer with a limit law, algebraic equivalence, bound or explicit one-sided comparison.
Exam move
Practise limits in mixed groups rather than by one technique at a time. For each item, write four labels before calculating: target, direction, substitution result and chosen route. Keep a separate list of failures caused by unequal one-sided limits, oscillation and unbounded behaviour.
Redo any cancellation with the excluded input written beside it, and check every radical rationalisation by multiplying the conjugates back together. Finish graph questions by stating both one-sided limits and the point value as three separate facts.
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