MATH1061 Chap.1 Limits and Continuity
Limits and Continuity
Limits describe what a function approaches near an input, so the nearby behaviour can matter even when the function is missing or differently defined at the point itself. This chapter separates left-hand limits, right-hand limits, two-sided limits and function values. It then turns continuity into a three-part test: the value exists, the limit exists and the two agree.
Algebraic simplification, conjugates and dominant-term reasoning are presented as repairs for specific indeterminate forms rather than automatic rituals. A graph supplies a useful first model: trace the curve toward the target from each side, record the two approach heights and only then inspect the filled point. The same order works algebraically for piecewise definitions.
Direct substitution is a diagnostic rather than a universal solution. A finite value may finish the problem, a nonzero number divided by zero signals unbounded behaviour or a missing limit, and zero over zero asks for more structure. Every conclusion should state which evidence establishes it, including whether side limits match and whether the target lies in the function's domain.
What this chapter covers
- 01
Read approach behaviour from graphs and tables
- 02
Compare left-hand and right-hand limits
- 03
Distinguish a limit from a function value
- 04
Resolve removable forms by factorisation
- 05
Use conjugates for radical expressions
- 06
Test continuity with three explicit conditions
Worked example · free
Compare two one-sided limits
- 1Use the lower branch from the left: x + 1 approaches 3.
- 1Use the upper branch from the right: 5 - x approaches 3.
- 1Because the one-sided limits agree, the two-sided limit is 3.
- 1The defined value is also 3, so all continuity conditions hold.
Key terms
- Left-hand limit
- The value approached when inputs move toward a point through smaller values.
- Right-hand limit
- The value approached when inputs move toward a point through larger values.
- Removable discontinuity
- A missing or mismatched point that can be repaired by redefining one value.
- Indeterminate form
- A substitution result that does not by itself determine the limit.
- Vertical asymptote
- A vertical line approached when function values become unbounded near an input.
- Squeeze law
- A limit principle that traps one function between two others sharing a limit.
- Dominant term
- The term controlling comparative growth as the input becomes large in magnitude.
- Punctured neighbourhood
- Inputs close to a point while excluding the point itself from consideration.
Limits and Continuity FAQ
Can a limit exist when the function value does not?
Yes. A limit depends on values arbitrarily close to the input, not necessarily at that input. A graph can have an open point while both sides approach the same height. The limit then exists even though the function value is missing. Filling that point with the approach value repairs a removable discontinuity.
When should I factor before taking a limit?
Factor when direct substitution produces a zero-over-zero form and the numerator and denominator may share a vanishing factor. Cancel only after stating the restriction on the original expression. The simplified expression describes nearby inputs, which is exactly what the limit needs, even though it may not equal the original formula at the point.
How do I prove continuity at one point?
Check three statements in order: the function value is defined, the two-sided limit exists and that limit equals the function value. For a piecewise function, compute the two one-sided limits separately. Writing all three conditions prevents a common mistake where matching side limits are shown but the actual defined value is ignored.
What does a vertical asymptote change about the limit?
A vertical asymptote often produces unbounded one-sided behaviour. Record the sign on each side rather than writing a single infinity automatically. If both sides grow in the same unbounded direction, an extended limit can describe that behaviour; if their signs differ, the ordinary two-sided limit does not exist. A nearby sign table makes the distinction explicit.
Exam move
Use a four-column grid headed left approach, right approach, common limit and defined value. For every graph or piecewise formula, fill those columns before writing any conclusion. Practise factorisation and conjugate examples only after stating why substitution was inconclusive.
Build a mixed set containing graphs, tables, rational expressions, radicals and piecewise rules so that the representation does not reveal the method in advance. On rational examples, write the original domain before cancelling anything. On radical examples, multiply by the conjugate and track the new denominator. For large-input limits, divide by the dominant power and explain which terms vanish.
When checking a solution, ask whether a cancelled expression was used only for nearby inputs and whether the final continuity claim explicitly compares the limit with the function value. Redo mistakes by classifying them as a side-limit error, a domain error or an algebraic repair error; each category needs a different correction.
Finish by explaining one example aloud without using the word substitute until the approach behaviour has been described.
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