1 2.6 – Limits Involving Infinity

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

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

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

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

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 Examples Evaluate the limit and determine any asymptotes

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