1. Section 1.8 Limits

2. Consider the graph of f(θ) = sin(θ)/θ

3. Let’s fill in the following table We can say that the limit of f(θ) approaches 1 as θ approaches 0 from the right We write this as We can construct a similar table to show what happens as θ approaches 0 from the left θ0.50.40.30.20.10.05 sin(θ)/θ θ0.50.40.30.20.10.05 sin(θ)/θ0.9590.9740.9850.9930.9980.9995 θ-0.5-0.4-0.3-0.2-0.1-0.05 sin(θ)/θ0.9590.9740.9850.9930.9980.9995

4. So we get Now since we have we say that the limit exists and we write

5. A function f is defined on an interval around c, except perhaps at the point x=c. We define the limit of f(x) as x approaches c to be a number L, (if one exists) such that f(x) is as close to L as we want whenever x is sufficiently close to c (but x≠c). If L exists, we write

6. Note: If f(x) is continuous at c, than so the limit is just the value of the function at x = c

7. We define _______to be the number L, (if one exists) such that for every ε > 0 (as small as we want), there is a δ > 0 (sufficiently small) such that if |x – c| < δ and x ≠ c, then |f(x) – L| < ε. L + ε L - ε L c-δ c c+δ ε ε f(x)f(x)

8. Properties of Limits

9. Compute the following limits

10. Let’s take a look at the last one What happened when we plugged in 1 for x? When we get we have what’s called an indeterminate form Let’s see how we can solve it

11. Let’s look at the graph of Seems to be continuous at x = 1

12. When does a limit not exist? When Example

14. Limits at Infinity If f(x) gets sufficiently close to a number L when x gets sufficiently large, then we write Similarly, if f(x) approaches L when x is negative and has a sufficiently large absolute value, then we write The line y = L is called a horizontal asymptote

15. Limits at Infinity Let’s show the following function has a limit, and thus a horizontal asymptote. So we need to calculate

16. Examples

17. Formal Definition of Continuity The function f is continuous at x = c if f is defined at x = c and if