MTH6131 Chap.2 Limits of Sequences and Functions
Limits of Sequences and Functions
From sequences to functions
The lectures introduce limits with sequences. A sequence is really a function defined on the natural numbers, so 1, 1/2, 1/3 and so on is the function s(n) = 1/n. When the terms approach a number L as n grows, L is the limit and the sequence converges. The limit need not be one of the terms: every term of 1/n is positive, yet the limit is zero.
Some sequences have no limit at all, because their terms keep alternating between two different values.
The definition that makes the idea precise
Intuition gets you started, but the lecture insists on the rigorous version: for every positive tolerance ε there is a natural number N such that every term from the N-th onward lies within ε of L.
The lecture reduces it to three components: the closeness condition, the tail from N onward, and the order of choice, tolerance first and N second.
The same structure appears for functions, where the tail is replaced by a small interval around the point: for every ε there is a δ such that f(x) lies within ε of L whenever x is within δ of a but not equal to it.
Two sides that must agree
Because x can approach a from the left or from the right, a function has two one-sided limits at each point, and the two-sided limit exists only when they agree.
The lecture's example 1/x at zero fails because the two sides run off in opposite directions, and piecewise and absolute value formulas are the usual places where the two sides give different finite values.
Theorems that let you compute
Limits respect sums, differences, constant multiples, products, quotients with a non-zero denominator, and roots.
Almost every calculation at this level turns on the quotient condition: when the denominator tends to zero together with the numerator, the expression has to be rewritten, by factorising, cancelling or expanding, before the theorem applies. The chapter closes on the trigonometric limit sin x over x, which equals 1 and underpins the derivative of sine.
What this chapter covers
- 01
Sequences as functions of a natural number
- 02
The epsilon and N definition of convergence
- 03
Infinity as a direction, not a number
- 04
The epsilon and delta definition for functions
- 05
Left and right limits, and when they disagree
- 06
The six theorems on limits
- 07
Rewriting a zero over zero quotient
- 08
Limits of sine and cosine
Worked example · free
A quotient that substitution cannot finish
- 1Substitute and identify the form.
- 2Factor numerator and denominator and cancel the common factor.
- 2Apply the quotient theorem to the reduced expression.
Key terms
- Convergent Sequence
- A sequence whose terms approach a single number as the index grows, so that every term far enough along lies within any chosen tolerance of it.
- Epsilon
- The tolerance in the rigorous definition of a limit, a positive number chosen first and as small as anyone likes.
- Left-hand Limit
- The value approached by f(x) as x increases towards a from below.
- Two-sided Limit
- The common value of the left-hand and right-hand limits, which exists only when both exist and agree.
- Zero over Zero
- The result of substitution in a quotient whose numerator and denominator both tend to zero, which calls for rewriting before any limit theorem is applied.
Limits of Sequences and Functions FAQ
Does a sequence have to reach its limit?
No. The terms only have to get arbitrarily close to it. The sequence 1/n never equals zero, yet zero is its limit, because for any tolerance the terms are eventually all within it. The lecture makes this point directly: the limit of a convergent sequence need not appear among its terms.
Why does the limit of 1/x as x approaches zero not exist?
Because the one-sided limits disagree. From the right the values grow without bound, and from the left they fall without bound. A two-sided limit requires both sides to approach the same finite number, so there is no limit, and writing infinity would hide the fact that the two sides go opposite ways.
Why is 0.999 recurring equal to 1?
The recurring decimal means the limit of the sequence 0.9, 0.99, 0.999 and so on. Its n-th term is 1 minus 10 to the power minus n, and since 10 to the power minus n tends to zero, the limit is exactly 1. A number defined as a limit equals that limit, so the two notations name the same number.
Exam move
Practise stating the epsilon and N definition from memory, then apply it to one sequence with a specific tolerance by finding a working N. Doing this twice, with different tolerances, fixes the order of choice better than rereading. For function limits, compute both one-sided limits every time a formula changes form at the point.
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