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

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

• Rates of change are a big part of calculus and where limits are needed

  • The average rate of change of a function $f$ between the points $x$ and $a$ is $\frac{f\left(x\right)-f\left(a\right)}{x-a}$

  • The rate of change at the instant $x=a$ cannot be found by substituting $x=a$ into the expression

    • $\frac{f\left(a\right)-f\left(a\right)}{a-a}=\frac{0}{0}$ which is undefined

  • Therefore, limits are used to find the behavior of the expression as $x$ gets closer to $a$

• A special limit notation is used to indicate this

  • For example $\underset{x\to c}{\lim }f\left(x\right)$ 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 $\underset{x\to c}{\lim }f\left(x\right)=R$ (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 behavior 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

• $\underset{x\to c^{-}}{\lim }f\left(x\right)$ 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

• $\underset{x\to c^{+}}{\lim }f\left(x\right)$ 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

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 $\underset{x\to c}{\lim }f\left(x\right)$

• 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

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 $f\left(x\right)=\frac{1}{x}$

  • $\underset{x\to 0}{\lim }f\left(x\right)$ 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 $g\left(x\right)=\cos \left(\frac{1}{x}\right)$

  • $\underset{x\to 0}{\lim }g\left(x\right)$ 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 $h\left(x\right)=\{\begin{array}{l}0,x<0\\ \frac{1}{2},x=0\\ 1,x>0\end{array}$

  • $\underset{x\to 0}{\lim }h\left(x\right)$ does not exist because the limit from the left does not equal the limit from the right at 0

    • $\underset{x\to 0^{-}}{\lim }h\left(x\right)=0$ but $\underset{x\to 0^{+}}{\lim }h\left(x\right)=1$

    • It doesn't matter that $h\left(0\right)$ has a well-defined value!