2LIMITS OF FUNCTIONS OBJECTIVES: define limits; illustrate limits and its theorems; andevaluate limits applying the given theorems.define one-sided limitsillustrate one-sided limitsinvestigate the limit if it exist or not using the concept of one-sided limits.define limits at infinity;illustrate the limits at infinity; anddetermine the horizontal asymptote.
3DEFINITION: LIMITS The most basic use of limits is to describe how a function behaves as the independent variableapproaches a given value. For example let us examine the behavior of the functionfor x-values closer and closer to 2. It is evident from the graph and the table in the next slide that the values of f(x) get closer and closer to 3 as the values of x are selected closer and closer to 2 on either the left or right side of 2. We describe this by saying that the “limit of is 3 as x approaches 2 from either side,” we write
4yf(x)3f(x)xO2x1.91.951.991.9951.99922.0012.0052.012.052.1F(x)2.712.8522.972.9852.9973.0033.0153.0313.1523.31left sideright side
5This leads us to the following general idea. (p. 70) Limits (An Informal View)
6EXAMPLEUse numerical evidence to make a conjecture about the value ofAlthough the function is undefined at x=1, this has no bearing on the limit.The table shows sample x-values approaching 1 from the left side and from the right side. In both cases the corresponding values of f(x) appear to get closer and closer to 2, and hence we conjecture thatand is consistent with the graph of f.
8THEOREMS ON LIMITSOur strategy for finding limits algebraically has two parts:First we will obtain the limits of some simpler functionThen we will develop a list of theorems that will enable us to use the limits of simple functions as building blocks for finding limits of more complicated functions.
9We start with the following basic theorems, which are illustrated in Fig 1.2.1 Theorem (p. 80)
12The following theorem will be our basic tool for finding limits algebraically. Theorem (p. 81)
13This theorem can be stated informally as follows: The limit of a sum is the sum of the limits.The limit of a difference is the difference of the limits.The limits of a product is the product of the limits.The limits of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.The limit of the nth root is the nth root of the limit.A constant factor can be moved through a limit symbol.
16ORWhen evaluating the limit of a function at a given value, simply replace the variable by the indicated limit then solve for the value of the function:
17Evaluate the following limits. EXAMPLE:Evaluate the following limits.Solution:(indeterminate)Equivalent function:
18Note: In evaluating a limit of a quotient which reduces to , simplify the fraction. Just remove the common factor in the numerator and denominator which makes the quotient .To do this use factoring or rationalizing the numerator or denominator, wherever the radical is.
19Rationalizing the numerator: Solution:(indeterminate)Rationalizing the numerator:
22DEFINITION: One-Sided Limits The limit of a function is called two-sided limit if it requires the values of f(x) to get closer and closer to a number as the values of x are taken from either side of x=a. However some functions exhibit different behaviors on the two sides of an x-value a in which case it is necessary to distinguish whether the values of x near a are on the left side or on the right side of a for purposes of investigating limiting behavior.
23Consider the function As x approaches 0 from the right, the values of f(x) approach a limit of 1, andsimilarly , as x approaches 0 from theleft, the values of f(x) approach alimit of -1.1-1
24This leads to the general idea of a one-sided limit (p. 72) One-Sided Limits (An Informal View)
251.1.3 (p. 73) The Relationship Between One-Sided and Two-Sided Limits
26EXAMPLE:1. Find if the two sided limits exist givenSOLUTION1-1
27EXAMPLE: 2. For the functions in Fig 1.1.13, find the one-sided limit and the two-sided limits at x=a if they exists.SOLUTIONThe functions in all three figures have the sameone-sided limits as , since the functions areIdentical, except at x=a.In all three cases the two-sided limit does not exist asbecause the one sided limits are not equal.
34DEFINITION: LIMITS AT INFINITY If the values of the variable x increase without bound, then we write , and if the values of x decrease without bound, then we writeThe behavior of a function as x increases or decreases without bound is sometimes called the end behavior of the function.For example ,