In this section, we will investigate the idea of the limit of a function and what it means to say a function is or is not continuous.

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Presentation transcript:

In this section, we will investigate the idea of the limit of a function and what it means to say a function is or is not continuous.

The limit of f as x approaches a is L means that f(x) can be made arbitrarily close to L by choosing x values sufficiently close to a. That is, the closer x gets to a (from either side), the closer y will get to L. Note: f(a) does not have to exist, and if it does, it is not necessarily true that f(a) = L.

Consider the graph below of y = f(x).

Consider the function.

xf(x) xf(x)

Consider the function. xf(x) xf(x)

Consider the function.

xf(x) xf(x)

Consider the function. xf(x) xf(x)

The right hand limit of f as x approaches a is L means that f(x) can be made arbitrarily close to L by choosing x values sufficiently close to, but larger than, x = a. That is, the closer x gets to a (from from the right), the closer y will get to L. Note: f(a) does not have to exist, and if it does, it is not necessarily true that f(a) = L.

The left hand limit of f as x approaches a is L means that f(x) can be made arbitrarily close to L by choosing x values sufficiently close to, but smaller than, x = a. That is, the closer x gets to a (from from the left), the closer y will get to L. Note: f(a) does not have to exist, and if it does, it is not necessarily true that f(a) = L.

Consider the graph below of y = f(x).

Consider the function. xf(x) xf(x)

Consider the function. xf(x) xf(x)

if and only if and So both left and right hand limits must agree for the overall limit to have a value.

Use the graph of y = f(x) below to find:

Algebraically find

Use the definition of derivative to find for the function

A function f is continuous at x = a if. So both the left and right hand limits must exist, the function itself must exist, and all three of these must be equal to each other.

Where is f not continuous? Why?

Consider the function Is f continuous at x = 4? Support your answer.

Find all values of “a” so that is continuous for all values of x.