2.6 – Limits involving Infinity, Asymptotes Math 1304 Calculus I 2.6 – Limits involving Infinity, Asymptotes

Recall: Limits of Infinity

Recall: Vertical Asymptotes Definition: The line x = a is called a vertical asymptote of the curve y = f(x) if any of the following limits exist

Limits as x approaches ±infinity Now consider:

Basic Examples f(x) = 1/x f(x) = 1/(x-a) f(x) = 1/x2 f(x)= 1/(x-a)2

Def of Limit as x approaches infinity Definition: Let f be a function defined on some interval (a,). We say that the limit as x approaches infinity is L if for any >0 there is a number N such that |f(x)-L|<  whenever x > N. In this case we write

Def of Limit as x approaches -infinity Definition: Let f be a function defined on some interval (-,a). We say that the limit as x approaches minus infinity is L if for any >0 there is a number N such that |f(x)-L|<  whenever x < N. In this case we write

Def of Limit of infinity as x approaches infinity Definition: Let f be a function defined on some interval (a,). We say that the limit as x approaches infinity is infinity if for any M there is a number N(M) such that f(x)>M whenever x > N(M). In this case we write

Horizontal Asymptotes Definition: The line y = L is called a horizontal asymptote of the curve y = f(x) if either of the following limits exist

Computational Methods for limits as x approaches +-infinity Rules Basic functions: constants, 1/x, 1/xr Sum, difference, product, quotient, power, etc. Algebraic Techniques For quotient of polynomials, divide by highest power (example next)

Basic Functions

Examples Find all horizontal and vertical asymptotes of the curve y=(2x-1)/(5x+3) y=(2x2-1)/(3x2+3x+6)