Section 1.8 Limits

Consider the graph of f(θ) = sin(θ)/θ

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 θ sin(θ)/θ θ sin(θ)/θ θ sin(θ)/θ

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

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

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

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)

Properties of Limits

Compute the following limits

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

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

When does a limit not exist? When Example

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

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

Examples

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