MATH1131 Chap.7 Limits and Continuity
Limits and Continuity
The calculus strand starts with limits. A limit as x tends to infinity describes long-run behaviour and has a precise definition: for every tolerance epsilon there is a threshold beyond which the function stays within epsilon of the limit.
Limit rules handle sums, products and quotients, rational functions are tamed by dividing through by the dominant power, differences of surds by multiplying by a conjugate, and oscillating expressions by the pinching theorem, which squeezes a function between two others with a common limit.
Limits at a point use left-hand and right-hand limits, which must agree for the limit to exist, and this matters for functions defined piecewise. Continuity at a point means the limit equals the function value, and continuous functions are closed under the usual combinations.
On a closed, bounded interval continuous functions satisfy two existence theorems: the intermediate value theorem, which guarantees roots between sign changes and underlies the bisection method, and the maximum-minimum theorem, which guarantees that largest and smallest values are attained.
What this chapter covers
- 01
Limits at infinity and dominant terms
- 02
The epsilon definition of a limit
- 03
The pinching theorem
- 04
One-sided limits and limits at a point
- 05
Continuity and piecewise functions
- 06
The intermediate value theorem and bisection
- 07
The maximum-minimum theorem
Worked example · free
A rational limit at infinity
- 1Divide every term by x^2, the largest power in the denominator: (4 - 3/x + 1/x^2)/(2 + 5/x^2).
- 1As x tends to infinity, 3/x, 1/x^2 and 5/x^2 all tend to 0.
- 1By the limit rules the expression tends to 4/2 = 2, since the denominator limit is non-zero.
Key terms
- Pinching theorem
- A result stating that a function squeezed between two others with the same limit has that limit too.
- One-sided limit
- The value a function approaches as x tends to a point from only the left or only the right.
- Continuous function
- A function whose limit at each point of its domain equals its value there.
- Intermediate value theorem
- The statement that a continuous function on a closed interval takes every value between its endpoint values.
- Bisection method
- A root-finding procedure that repeatedly halves an interval on which a continuous function changes sign.
- Jump discontinuity
- A point where both one-sided limits exist but are different.
Limits and Continuity FAQ
How do I prove a limit using the pinching theorem?
Find two simpler functions that bound yours above and below on the relevant range, justify both inequalities, and show the bounds share the same limit. Then quote the theorem by name to conclude.
Why might a limit at a point not exist even if the function is defined there?
The limit depends on values near the point, not at it. If the left-hand and right-hand limits differ, or the function oscillates without settling, the limit does not exist regardless of the function value at that point.
What conditions does the intermediate value theorem need?
Continuity of the function over the whole closed interval [a, b] is required. Then each value between f(a) and f(b) is attained at some point of the interval. Without continuity a sign change does not guarantee a root.
How do I choose k to make a piecewise function continuous?
Compute the left-hand limit, the right-hand limit and the function value at the join, each from the correct formula, and set them equal. Solve the resulting equation for k.
Does the maximum-minimum theorem work on open intervals?
Not in general. On an open interval a continuous function may approach a value without reaching it, as 1/x does on (0, 1), which has no maximum. The theorem needs a closed, bounded interval.
Why do I divide by the largest power in the denominator?
Dividing every term by that power turns the denominator into a quantity with a non-zero limit, while lower powers become terms that vanish. The limit rules can then be applied safely to the simplified expression.
Exam move
Sort practice limits by type before solving them: polynomial ratio, surd difference, bounded oscillation, or one-sided at a point. Writing the type first trains the recognition step that the exam rewards. For theorem questions, memorise the exact hypotheses of the intermediate value and maximum-minimum theorems and practise writing a two-line justification that checks them.
Use counterexamples, such as a jump function or 1/x on an open interval, to understand why each hypothesis is needed. Reserve time for epsilon arguments, which appear in the notes and test precise reading of quantifiers. Keep a short list of standard limits, such as sin x over x tending to one at zero, and practise recognising when a new limit reduces to one of them after a substitution.
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