Limits of Functions

In physics, relationships between quantities are expressed with functions using constants and variables.

Function: a relationship between variables, usually an equation or a graph. In an explicit function one variable is expressed in terms of the other. Example: y = x (tan 60^o), often written as:  f (x) = x(tan 60^o)  In an implicit function the variables are mixed together, unsolved. Example: y/x =tan 60^o A function is continuous if the function is defined for all values of the independent variable.

Variable: a quantity that can change with the circumstances and may assume an indefinite number of values. Usually represented by letters s – z. The variable is continuous if it can assume all values between upper and lower boundaries.

Constant: a term that has a fixed value, either absolute (p, e, c) or arbitrary.

Independent variable: a quantity that is controlled by the experimenter in order to observe the change in a related variable.

Dependent variable: one whose value depends on the value of another variable. The dependent variable is a function of the independent variable.

Limit of a Sequence

Consider the following sequence of numbers: 1, 3/2, 5/3, 7/4, 9/5, …, 2 – 1/n, … As the sequence is continued to larger numbers of n, the value of the sequence approaches 2 but never actually reaches it. We say it approaches 2 as a limit and write it as .

Limit of a Function

Consider the function f (x) = x² .  Now if , as in the sequence above, the corresponding values of f (x) would be 1, 9/4, 25/9, 49/16, …, (2-1/n)², … If the sequence is continued, the value of  f(x) approaches 4 or . But x could approach the value of 2 from above as well as below. A quick examination of the sequence of x values 2.1, 2.01, 2.001, 2.001 … substituted into the function f (x) = x² shows again that as  , . We say “the limit, as x approaches 2, of x² is 4” and write _.

Right and Left Limits

In the above example we see that as x approaches 2 from the left (on a number scale) the function value approaches but is always less than 4. As x approaches 2 from the right, the function value approaches but is always greater than 4. The limits from the left and right side are the same in this case and no distinction is necessary, but this is not always the case. The existence of a right or left limit does not necessarily imply the existence of the other limit. To indicate the left limit we write:  and for the right limit:

Theorems on Limits

1. If f(x) = c, a constant, then

If    and    then:

2. _, k being any constant
3. 
4. 
5.   provided B is not zero
6.   provided the result is a real number

Infinity

A variable or a function is said to become positively infinite if at some point in a sequence or limit it becomes and remains greater than any preassigned positive number, however large.

A variable or a function is said to become negatively infinite if at some point in a sequence or limit it becomes and remains less than any preassigned negative number, however small.

Often we are concerned only with the absolute value, in which case the function or variable is said to become infinite, without regard to sign.

Examples of infinite limits:

          

Do not use the theorems for multiplication and division above with functions having infinite limits, but only for functions whose limits exist.

Continuity

A function f (x) is continuous at x = a if:

1. f (a) is defined
2.  exists
3. 

Examples:

(a.)  is discontinuous at x = 2 since f (2) is not defined and the limit at x = 2 does not exist (it is infinite). It is continuous everywhere except at x = 2 where it is said to have an infinite discontinuity.

 

(b.)  is discontinuous at x = 2 since f (2) is not defined (both numerator and denominator are zero).  The limit exists and is equal to 4, but is not equal to the value of f (2). This discontinuity is called removable since it may be removed by redefining the function as . Factoring and simplifying the function gives the function g (x) = x + 2 which has the same graph as f (x) except for at x = 2 where the graph of f (x) has a hole.

 

(c.) A function that changes over a range of values will exhibit jump discontinuities.

The following function: If 0 < x > 3,  f (x) = 5; if x > 3, f (x) = 7 will have a jump discontinuity at x = 3. The left limit does not equal the right limit so the limit of the function at x = 3 does not exist.