1. 1 2.6 – Limits Involving Infinity

2. 2 Definition The notation means that the values of f (x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a (on either side) but not equal to a. a f

3. 3 Vertical Asymptote The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following six statements is true:

4. 4 Examples Find the limit. 1. 2.State all vertical asymptotes for the following function and write the equivalent limit statement for each asymptote.

5. 5 Definition Let f be a function defined on some interval (a, ∞). Then means that the value of f (x) can be made as close to L as we like by taking x sufficiently large. L f

6. 6 Horizontal Asymptote The line y = L is called a horizontal asymptote of the curve y = f(x) if either or

7. 7 Examples Evaluate the following. State the equations of any asymptotes that result from the limit.

8. 8 Algebra Review 1.Simplify 2. Bring the expression into the radical and simplify.

9. 9 Properties If n is a positive integer, then, where a is some constant. To evaluate limits going to infinity, we often use the technique of multiplying the expression by 1 in the form of

10. 10 Examples Evaluate the limit and determine any asymptotes. 1.2. 3.

11. 11 You Try It Evaluate the limit and use the results to state any asymptotes that exist. 1.2. 3.4.