Definition of a Limit (College Board AP® Calculus AB)

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Written by: Roger B

Reviewed by: Dan Finlay

What are limits in mathematics?

When we consider a limit in mathematics we look at the tendency of a mathematical process as it approaches, but never quite reaches, an ‘end point’ of some sort
- Most of the key ideas in calculus rely on limits for their basic definitions
- In this course you will be usually considering the limits of functions
A special limit notation is used to indicate this
- For example $\lim_{x \to c} f (x)$ denotes ‘the limit of the function f as x goes to (or approaches) c’
  - I.e., what value (if any) f(x) gets closer and closer to as x takes on values closer and closer to c
    - If the limit exists, we write $\lim_{x \to c} f (x) = R$ (where $R$ is a real number)
    - In that case the value of f(x) can be made as close as you want to R, by making x sufficiently close to (but not equal to) c
  - We are not concerned here with what value (if any) f(x) takes when x is equal to c
    - We only care about the behaviour of f(x) as x gets close to c

One-sided limit

What are one-sided limits?

A one-sided limit is only concerned with the value of the function as you approach a particular value of x from one side or the other
$\lim_{x \to c^{-}} f (x)$ indicates the one-sided limit from the left (or from below)
- It is the value that f(x) approaches as x gets closer and closer to c from the x<c side
$\lim_{x \to c^{+}} f (x)$ indicates the one-sided limit from the right (or from above)
- It is the value that f(x) approaches as x gets closer and closer to c from the x>c side

Worked Example

The figure below shows the graph of the function $f$ defined by $f (x) = \{\begin{cases} x + 1, x \leq 2 \\ 4, x > 2 \end{cases}$ .

A graph of y=x+1 for x<=2, and y=4 for x>2

Find:

(a) $f (2)$

Answer:

This is simply the value of the function when $x = 2$

$f (2) = 3$

(b) $\lim_{x \to 2^{-}} f (x)$

Answer:

This is the one-sided limit from the left

From the graph, or from considering the function algebraically, we can see that the value of the function gets closer and closer to 3 as $x$ gets close to 2 from the left

$\lim_{x \to 2^{-}} f (x) = 3$

Answer:

This is the one-sided limit from the right

From the graph, or from considering the function algebraically, we can see that the value of the function is always equal to 4 as $x$ gets closer and closer to 2 from the right

Note that it doesn't matter what the value of the function is at $x = 2$

$\lim_{x \to 2^{+}} f (x) = 4$

Two-sided limit

What are two-sided limits?

The two-sided limit (or simply the limit) of a function f at the value x=c is denoted by $\lim_{x \to c} f (x)$
To find a two-sided limit, consider the one-sided limits from the left and the right
- If the two one-sided limits are equal, then that is the value of the two-sided limit
- If the two one-sided limits are not equal, then the two-sided limit for that x value doesn't exist

Worked Example

The figure below shows the graph of the function $f$ defined by $f (x) = \{\begin{cases} x^{2} + 1, x \neq 0 \\ 0, x = 0 \end{cases}$ .

Graph of y=x^2+1 for x not equal to 0, and y=0 for x=0

Find:

(a) $f (0)$

Answer:

This is simply the value of the function when $x = 0$

$f (0) = 0$

(b) $\lim_{x \to 0} f (x)$

Answer:

The limit does not care what the value at $x = 0$ is; it only cares what happens as $x$ gets close to zero

From the graph, or from considering the function algebraically, we can see that the value of the function gets closer and closer to 1 as $x$ gets close to zero

$\lim_{x \to 0} f (x) = 1$

Nonexistent limits

When does the limit of a function at a point not exist?

For some functions, a limit might not exist for certain values of x
- We say that the limit is nonexistent for such cases
There are three common ways that a limit may not exist at a point:

Case 1: The function is unbounded near the point in question

E.g., consider the function $f$ defined by $f (x) = \frac{1}{x}$
- $\lim_{x \to 0} f (x)$ does not exist because the function becomes unbounded as x approaches 0
  - It is unbounded in the positive direction as x approaches 0 from the right
  - And in the negative direction as x approaches 0 from the left

Case 2: The function oscillates near the point in question

E.g., consider the function $g$ defined by $g (x) = \cos (\frac{1}{x})$
- $\lim_{x \to 0} g (x)$ does not exist because the function oscillates as x approaches 0
  - The function is not unbounded near zero (its values are always between -1 and 1)
  - But it oscillates rapidly between -1 and 1 and never 'settles down' to a single value

Case 3: The one-sided limits are not equal at the point in question

E.g., consider the function $h$ defined by $h (x) = \{\begin{cases} 0, x < 0 \\ \frac{1}{2}, x = 0 \\ 1, x > 0 \end{cases}$
- $\lim_{x \to 0} h (x)$ does not exist because the limit from the left does not equal the limit from the right at 0
  - $\lim_{x \to 0^{-}} h (x) = 0$ but $\lim_{x \to 0^{+}} h (x) = 1$
  - It doesn't matter that $h (0)$ has a well-defined value!

Worked Example

Explain why each of the following limits is nonexistent:

(a) $\lim_{x \to - 1} \sin (\frac{1}{x^{2} - 1})$

Answer:

Note that when $x = - 1$ , $x^{2} - 1 = 0$

The function is undefined at $x = - 1$ , but that isn't what makes the limit non-existent; rather it's the way the function starts to oscillate as $x$ gets close to $- 1 :$

A graph of y=sin(1/(x^2-1)) from x=-2 to x=2

As x gets close to -1, x²-1 gets close to zero. Therefore the function oscillates between -1 and 1 as x approaches -1, which means the limit does not exist.

(b) $\lim_{x \to \frac{π}{2}} \tan x$

Answer:

Remember what happens to the graph of $\tan x$ as $x$ gets near to $\frac{π}{2}$ :

A graph of y=tanx between -π and π, showing asymptotes at -π/2 and π/2

tanx becomes unbounded as x approaches π/2 from the left and right. Therefore the limit does not exist.

(c) $\lim_{x \to 0} f (x)$ , where $f$ is the function defined by $f (x) = \{\begin{cases} - π, x < 0 \\ π, x \geq 0 \end{cases}$

Answer:

The graph of the function can show the problem with the limit here:

A graph of y=-π for x<0 and y=π for x>=0

$\lim_{x \to 0^{-}} f (x) = - π$

$\lim_{x \to 0^{+}} f (x) = π$

The limits from the left and right at x=0 are not equal, therefore the limit does not exist.

Last updated: 1 August 2024

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Definition of a Limit (College Board AP® Calculus AB)

Study Guide

What are limits in mathematics?

One-sided limit

What are one-sided limits?

Worked Example

Two-sided limit

What are two-sided limits?

Worked Example

Nonexistent limits

When does the limit of a function at a point not exist?

Case 1: The function is unbounded near the point in question

Case 2: The function oscillates near the point in question

Case 3: The one-sided limits are not equal at the point in question

Worked Example

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Author: Roger B

Author: Dan Finlay