Limits and Continuity

Calculus relies on the principle of using approximations of increasing accuracy to find the exact solution. This principle is applied to its building blocks - functions between sets of real numbers - using the concept of a limit. Knowledge of how functions behave as the variable comes arbitrarily close to a certain point, or goes off to +/- infinity, is absolutely fundamental to the theory of calculus. The first thing we will use limits for is to define continuity, but, as we will see later, also finding the instantaneous rate of change of a function (differentiation), and calculating the area under a curve (integration) is defined by taking limits of something.

Introductory videos: teaser and thorough introduction.

Topics

(Click on the topics for details.)

- An informal definition of limit

- An informal definition of limit


Definition 1: An informal definition of limit
Let \(f\) be a function defined for all points near \(a\), except possibly at \(a\) itself. We say that \(L\) is the limit of \(f\) as \(x\) approaches \(a\), if \(f(x)\) can be made as close as desired to \(L\) by making \(x\) close enough, but not equal, to \(a\).

This is written as

\[L=\lim_{x\to a} \ f(x).\]

Relevant parts of the book: Sections 1.1-1.2
Relevant videos:
- Hva skal vi med grenseverdier? (08:01-11:52)
- Hva skal vi med grenseverdier? (00:00)-(02:18).
Relevant examples:
- The limit does not depend on the value of the function at the limit point
Relevant Maple worksheets:
- Basic limits
- Special examples
Relevant exercises: 1.1.5-8, 1.1.12-13, 1.2.1


- Left and right limits

- Left and right limits


Sometimes we are interested in the behavior of a function only at one side of a point, like if the point is an endpoint of an interval we consider. We therefore introduce the concepts of left and right limits.

Definition 2: An informal definition of left and right limits
We say that \(L\) is the left limit of the function \(f\) at a point \(a\) if we can get \(f(x)\) as close as we want to \(L\) by taking \(x\) to the left of \(a\) and close to \(a\), but not equal to \(a\).

We write \[\lim_{x\to a^-}f(x)=L\]

The right limit is defined similarly

Theorem 1: Relationship between one-sided and two-sided limits
A function \(f\) has limit \(L\) as \(x\) approaches \(a\) if and only if \(f(x)\) has both a left and a right limit as \(x\) approaches \(a\) and these one-sided limits both equal \(L\).

That is: \(\lim_{x\to a}f(x)=L\iff \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\).

Relevant parts of the book: Section 1.2
Relevant examples:
- A function that has different left and right limits
Relevant Maple worksheets:
- Basic limits
- Special examples
Relevant exercises: 1.2.3, 1.2.5, 1.2.55, 1.2.57


- Rules for computing limits

- Rules for computing limits


There are rules for calculating limits that are useful. When the limits exist, one may add, subtract and multiply limits just as one can with numbers. Most of the following rules for calculating limits are fairly straightforward from the definition, but still wise to remember.

Theorem 2: Limit rules
Suppose \(\lim_{x\to a}f(x)=L\), \(\lim_{x\to a}g(x)=M\) and \(k\) is a constant. Then the following rules for computing limits hold

  1. \(\lim_{x\to a}[f(x)+g(x)]=L+M\)
  2. \(\lim_{x\to a}[f(x)-g(x)]=L-M\)
  3. \(\lim_{x\to a}f(x)g(x)=LM\)
  4. \(\lim_{x\to a}kf(x)=kL\)
  5. \(\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{L}{M}, \quad \text{if} \,\, M\neq 0.\)
  6. Furthermore, if \(m\) is an integer and \(n\) is a positive integer, then \(\lim_{x\to a}[f(x)]^{m/n}=L^{m/n}\), if \(L>0\) when \(n\) is even, and \(L\neq 0\) if \(m<0\).
  7. If there is a \(c>0\) such that \(f(x)\le g(x)\) for all \(x\in (a-c,a)\cup (a,a+c)\), then \(L\le M\).

(Notice that we in 7. cannot conclude that \(L<M\) if \(f(x)< g(x)\) for all \(x\in (a-c,a)\cup (a,a+c)\).)

These rules also hold for left limits and right limits.

Another rule that is useful for computing limits is the Squeeze Theorem.

Theorem 4: The Squeeze Theorem
If there is a \(c>0\) such that \(f(x)\le g(x)\le h(x)\) for all \(x\in (a-c,a)\cup (a,a+c)\), and \(\lim_{x\to a}f(x)=\lim_{x\to a}h(x)=L\), then \(\lim_{x\to a}g(x)=L\).

This rule also holds for left limits and right limits.

Relevant parts of the book: Section 1.2
Relevant examples:
- Computing limits
- Limits of polynomials and rational functions
- Using the Squeeze Theorem
Relevant Maple worksheets:
- Basic limits
- Special examples
Relevant exercises: 1.2.9, 1.2.23, 1.2.25, 1.2.75, 1.2.77


- Limits at infinity

- Limits at infinity


Sometimes we are interested in the "long term" behaviour of a function. That is, the eventual behaviour as the variable increases/decreases towards +/- infinity. The following definition is useful for that purpose.

Definition 3: An informal definition of limits at infinity and negative infinity
We say that \(L\) is the limit of \(f\) at infinity if \(f\) can be made as close as desired to \(L\) by taking \(x\) large enough.

This is written as \[\lim_{x\to\infty}f(x)=L.\]

Limits at negative infinity are defined similarly, then with the requirement that \(f(x)\) can be made as close as desired to \(L\) by taking \(-x\) large enough. This is written as \[\lim_{x\to -\infty}f(x)=L.\]

The above rules for computing limits are also valid for \(\lim_{x\to\pm\infty}f(x)\).

Limits at \(\pm\infty\) are used to define horizontal asymptotes.

Relevant parts of the book: Section 1.3
Relevant videos:
- Hva skal vi med grenseverdier? (05:31-08:00).
Relevant examples:
- Limits at infinity
- Using the Squeeze Theorem to compute limits at infinity
- Limits at infinity for rational functions
Relevant Maple worksheets:
- Basic limits
- Special examples
Relevant exercises: 1.3.3, 1.3.29, 1.3.33


- Infinite limits

- Infinite limits


A function \(f\) is said to approach infinity as \(x\) approaches \(a\) if \(f(x)\) can be made as large as desired by making \(x\) close enough, but not equal, to \(a\).

We write \(\lim_{x\to a}f(x)=\infty\). Note that in writing this we are not saying that \(\lim_{x\to a}f(x)\) exists, rather we are saying that \(\lim_{x\to a}f(x)\) does not exist because \(f(x)\) becomes arbitrarily large near \(x=a\).

\(\lim_{x\to a}f(x)=-\infty\), \(\lim_{x\to a^-}f(x)=\pm\infty\), \(\lim_{x\to a^+}f(x)=\pm\infty\) and \(\lim_{x\to\pm\infty}f(x)=\pm\infty\) are defined in similar ways.

Infinite limits are used to define vertical asymptotes.

Relevant parts of the book: Section 1.3
Relevant examples:
- Infinite limits
Relevant Maple worksheets:
- Basic limits
- Special examples
Relevant exercises: 1.3.13, 1.3.33


- Continuity

- Continuity


When taking limits and in the applications mentioned above, it is clearly desirable that the function does not suddenly make great changes arbitrarily close to the point. This introduces the concept of a continuous function. Informally, a function \(f\) is continuous at a point \(x\) if points close to \(x\) are mapped close to \(f(x)\), or equivalently if the graph of the function is an unbroken curve. However, we need a more formal definition than this to work with the concept in general.

Definition 4: Continuity at a point
The function \(f\) is continuous at some point \(a\) of its domain if the limit of \(f(x)\) as \(x\) approaches \(a\) exists and is equal to \(f(a)\). This is written as \[\lim_{x \to a} \ f(x) = f(a).\]

This is equivalent to the following: For every \(\varepsilon>0\), there exists a \(\delta>0\) such that \(|a-x|<\delta\) implies \(|f(a)-f(x)|<\varepsilon\).

We may also define left and right continuity of a function, in parallel with left and right limits defined above. This is necessary for defining continuity in an endpoint, where it may not make sense/be of interest to talk about how the function behaves approaching the point from outside the interval.

Definition 5: Left and right continuity, and continuity in endpoints
We say that a function \(f\) is left continuous at a point \(a\) if

\[ \lim_{x\to a^-} f(x)=f(a).\]

Right continuity is defined similarly. We say that a function defined on an interval is continuous in the endpoints if it is right continuous at the left endpoint, and vice versa.

Relevant parts of the book: Section 1.4
Relevant videos:
- Hva skal vi med grenseverdier (00:00-05:30) and (08:01-11:52)
- Hva skal vi med grenseverdier (00:00)-(02:18).
-Exam august 1, 2012 problem 2
Pencasts:
- Intuitive meaning of continuity (1:42)
Relevant examples:
- Continuous Extensions and Removable Discontinuities
Relevant Maple worksheets:
- Basic limits
- Special examples
Relevant exercises: 1.4.1, 1.4.3, 1.4.5, 1.4.11, 1.4.15, 1.4.17, 1.4.23


- The Max-Min Theorem and the Intermediate Value Theorem

- The Max-Min Theorem and the Intermediate Value Theorem


Why study continuity? The virtue of continuous functions largely stems from two major theorems. The Max-Min theorem guarantees the existence of a maximum and a minimum for continuous functions on closed intervals. The Intermediate Value Theorem says that that if the domain of a continuous function is an interval, then so is its range.

Theorem 8: The Max-Min Theorem
If \(f\) is continuous on the closed, finite interval \([a,b]\), then there exist numbers \(p\) and \(q\) in \([a,b]\) such that for all \(x\) in \([a,b]\) \[f(p) \le f(x) \le f(q).\]

Slogan form: "Continuous functions on closed intervals take on their extrema."

Theorem 9: The Intermediate Value Theorem
If a function \(f\) is a continuous on the interval \([a, b]\), and \(u\) is a number between \(f(a)\) and \(f(b)\), then there is a \( c ∈ [a, b]\) such that \(f(c) = u.\)

Slogan form: "The continuous image of an interval is also an interval."

Relevant parts of the book: Section 1.4
Pencasts:
- The Intermediate Value Theorem (1:41)
- The Max-Min Theorem (1:30)
Relevant examples:
- Using the Intermediate-Value Theorem to find zeros of functions
Relevant Maple worksheets:
- Basic limits
- Special examples
Relevant exercises: 1.4.29, 1.4.31


- The Formal Definition of Limit

- The Formal Definition of Limit


The informal definition of a limit given above is enough to get an intuitive understanding of limits, and in some cases compute them, but in general it is far to imprecise to work with. The following definition formalizes the above description of a limit.

Definition 8: A formal definition of limit
Let \(f\) be a function defined on an interval around the point \(a\) (except possibly at the point \(a\) itself, i.e. a punctured interval). We say that \[L=\lim_{x\to a} \ f(x)\] provided that, for every \(\epsilon\) >0, there exists a \(\delta > 0\) such that for all \(x\), \[0 < | x - a | < \delta \implies | f(x) - L | < \epsilon\]

Left and right limits, limits at infinity and negative infinity and infinite limits can be defined in similar ways.

Relevant parts of the book: Section 1.5
Relevant videos:
- Definisjon av grenseverdier.
Pencasts:
- Limit of (5-2x) as x tends to 2
Examples:
- Error tolerance
- Proving a theorem using the definition
Relevant Maple worksheets:
- Basic limits
- Special examples
Relevant exercises: 1.5.1, 1.5.5, 1.5.9, 1.5.20


2013-10-02, nikolau