1. 2.6 – Limits involving Infinity, Asymptotes
Math 1304 Calculus I
2.6 – Limits involving Infinity, Asymptotes

2. Recall: Limits of Infinity

3. 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

4. Limits as x approaches ±infinity
Now consider:

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

6. 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

7. 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

8. 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

9. 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

10. 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)

11. Basic Functions

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