2INFINITE LIMITS; VERTICAL AND HORIZONTAL ASYMPTOTES; SQUEEZE THEOREM OBJECTIVES:define infinite limits;illustrate the infinite limits ; anduse the theorems to evaluate the limits of functions.determine vertical and horizontal asymptotesdefine squeeze theorem
3DEFINITION: INFINITE LIMITS Sometimes one-sided or two-sided limits fail to exist because the value of the function increase or decrease without bound.For example, consider the behavior of forvalues of x near 0. It is evident from the table and graph in Fig that as x values are taken closer and closer to 0 from the right, the values ofare positive and increase without bound; and asx-values are taken closer and closer to 0 from the left, the values of are negative and decrease without bound.
4In symbols, we writeNote:The symbols here are not real numbers; they simply describe particular ways in which the limits fail to exist. Thus it is incorrect to write
14DEFINITION:The line is a vertical asymptote of the graph of the function if at least one of the following statement is true:
15The following figures illustrate the vertical asymptote . x=ax=a
16The following figures illustrate the vertical asymptote . x=ax=a
17DEFINITION:The line is a horizontal asymptote of the graph of the function if either
18The following figures illustrate the horizontal asymptote y=by=b
19The following figures illustrate the horizontal asymptote y=by=b
20Determine the horizontal and vertical asymptote of the function and sketch the graph.a. Vertical Asymptote:Equate the denominatorto zero to solve for thevertical asymptote.b. Horizontal Asymptote:Divide both the numerator and the denominator by the highest power of x to solve for the horizontal asymptote.Evaluate the limit as xapproaches 2
26LIMITS OF FUNCTIONS USING THE SQUEEZE PRINCIPLE The Squeeze Principle is used on limit problems where the usual algebraic methods (factoring, conjugation, algebraic manipulation, etc.) are not effective. However, it requires that you be able to ``squeeze'' your problem in between two other ``simpler'' functions whose limits are easily computable and equal. The use of the Squeeze Principle requires accurate analysis, algebra skills, and careful use of inequalities. The method of squeezing is used to prove that f(x)→L as x→c by “trapping or squeezing” f between two functions, g and h, whose limits as x→c are known with certainty to be L.