MATH1011 Chap.8 Limits, Asymptotes and Continuity
Limits, Asymptotes and Continuity
A limit answers a different question from a value
Ask for f(3) and you are asking for the one number the rule produces at 3. Ask for the limit of f(x) as x approaches 3 and you are asking a question about the neighbours of 3 only: what number are the outputs crowding towards, with x = 3 itself ruled out of the discussion.
The two answers usually agree, which is exactly why the distinction slips past people, and they are allowed to disagree in either direction.
A function can have no value at 3 and a perfectly good limit there, or a value at 3 that nothing around it goes anywhere near.
A two sided limit is assembled from two one sided ones: the left limit, taken as x climbs towards the target, and the right limit, taken as x comes down onto it. If both exist and match, that shared number is the limit. If they differ, no limit exists, however tidy the formula looks.
On a rule given in pieces, work each side with the branch that governs that side, and remember that the line assigning a height exactly at the join answers the value question, never the limit question.
Naming the misbehaviour, not just reporting it
When a limit does not exist, writing that it does not exist is only half an answer, because three quite different things can have gone wrong and the marks sit in saying which.
The two sides may settle on different finite heights, which is a jump, and piecewise rules are the usual source. The outputs may grow past every bound you care to name, which is the unbounded case, and here the infinity symbol records a behaviour rather than a number you are allowed to add or cancel.
Or the outputs may keep swinging and never settle at all, which is oscillation; the sine of 1/x near the origin is the standard illustration, and zooming in never calms it.
Keep those three apart from a fourth situation that is not a failure: the hole, where both sides agree on a height that the function simply does not take.
The hole is the only one of the four you can mend, and you mend it by defining the missing height to be the limit.
What to do when substitution jams
Always put the number straight in first, since that settles most of the limits in this course on its own. When it jams, the jam has a recognisable shape and each shape has one repair.
A quotient of polynomials returning zero over zero is telling you that the same factor sits on the top and on the bottom: factor both, cancel it, and substitute again.
Cancelling is legal here because x is only ever near the target and never at it, so nothing has been divided by 0. A zero over zero involving a square root difference clears when you multiply the top and the bottom by the same expression with its middle sign reversed, since a sum times a difference removes the root.
If x is running off to infinity instead, divide every term by the largest power of x in the bottom; each leftover then sits above a power of x and fades away, leaving the ratio of the leading coefficients.
Watch the sign of x at minus infinity, where odd powers change the result.
Tick three boxes at the point, then read the lines off
Continuity at x = a is not an impression of the graph, it is three checks in a fixed order. Does f(a) exist? Does the two sided limit at a exist? Are those two numbers equal? Run the cheap check first, because a missing value ends the argument on the spot.
Which check fails also names what you have found: no value but a sound limit is a hole, two sides disagreeing is a jump, unbounded sides are an infinite break.
Polynomials, sine, cosine and exponentials pass all three everywhere, and sums, products, quotients with a bottom that does not vanish, and compositions of continuous functions inherit the property, so the only points you ever have to argue about are the awkward ones.
Asymptotes need no separate theory, because each one is a limit you have already computed.
A vertical line x = a belongs on the sketch when at least one one sided limit at a is unbounded, which for a fraction happens where the bottom vanishes and the top does not; if both vanish you have a hole and no line at all. A horizontal line comes from the limits at plus and minus infinity, and the two ends are allowed to give different heights, so take both.
When the top of a fraction is exactly one degree above the bottom, the graph follows a slanted line instead, and dividing out reveals it as the straight part of the quotient.
What this chapter covers
- 01
Approaching a height against actually taking it
- 02
One sided limits, and what happens when they disagree
- 03
Factoring, cancelling and the conjugate for a jammed quotient
- 04
Behaviour when the input runs off to infinity
- 05
The three part check for continuity at a point
- 06
Vertical, horizontal and slanted asymptotes read from limits
Worked example · free
One fraction, one hole and one asymptote
- 2Factor both quadratics, cancel the shared factor x + 2, and substitute -2 into what is left.
- 2At x = 2 test the top and the bottom separately, then try one input just below 2 and one just above it to get the direction on each side.
- 2Divide every term by x² and read the ratio of the leading coefficients.
- 1Classify the two bad inputs and say which one a redefined height would fix.
Key terms
- One Sided Limit
- A one sided limit is the height the outputs approach when the input is allowed in from the left only, or from the right only. The two sided limit exists precisely when both one sided limits exist and are equal.
- Indeterminate Form
- An indeterminate form is a symbolic outcome such as zero over zero that substitution returns when it has not settled the limit. It is an instruction to rewrite the expression, never an answer in its own right.
- Removable Discontinuity
- A removable discontinuity is an input where the two sided limit exists but the function has no height there, or a height different from that limit. Redefining the single height to equal the limit repairs it.
- Jump Discontinuity
- A jump discontinuity is an input where the left and right limits both exist as finite numbers but differ from each other. No choice of height at that input can close the gap between them.
- Vertical Asymptote
- A vertical asymptote is a line x = a that the graph runs alongside because at least one one sided limit at a is unbounded. For a fraction it appears where the bottom vanishes and the top does not.
- Horizontal Asymptote
- A horizontal asymptote is a line y = L that the graph closes in on as the input runs off to plus or minus infinity. The two ends may produce different lines, so both limits have to be taken.
- Continuity At A Point
- Continuity at a point holds when the function has a height there, the two sided limit exists there, and those two numbers agree. Losing any one of the three breaks the graph at that input.
Limits, Asymptotes and Continuity FAQ
Can a function have a limit somewhere it is not even defined?
Yes, and that case turns up constantly. A limit inspects inputs near the target and never the target itself, so a fraction that reduces to something harmless once a shared factor cancels has a perfectly good limit at the very input it skips. The graph shows a hole at that spot.
What is an answer of zero over zero actually telling me?
It tells you that substitution was the wrong instrument, not that the limit is 0 and not that it fails to exist. Factor and cancel when the parts are polynomials, or multiply by the conjugate when a square root difference appears, then substitute into the rewritten expression.
How do I tell a hole from a vertical asymptote without sketching the graph?
Put the suspect input into the top and the bottom separately. A bottom of 0 with a nonzero top forces the values to grow without bound, which is an asymptote. A top and bottom that both vanish share a factor, and cancelling it usually leaves a finite limit, which is a hole.
Exam move
Collect about eight fractions and piecewise rules, then deliberately stop after two lines on each: write what substitution returned, and name what jammed. Sort them into the four outcomes this chapter keeps apart, a plain value, a hole, a jump and an unbounded break, before you finish a single one of them properly.
Most of the marks lost in this chapter are lost in that first diagnosis rather than in the algebra that comes after it.
Working through Limits, Asymptotes and Continuity in MATH1011? Sia is AskSia’s AI Mathematics tutor — ask any MATH1011 Limits, Asymptotes and Continuity question and get a clear, step-by-step explanation grounded in how MATH1011 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
