Text description - screencast for exercise 7

[The narrator reads out the onscreen text.]

NARRATOR: Exercise 7. Examine whether or not the function f of x equals 4 minus x squared if x is less than or equal to 0 or 4 plus x if x is greater than 0 is continuous at x equals 0. We first consider the limit as x approaches 0 from both above and below. As x approaches 0 from below, we'll be dealing with the part of the function with rule 4 minus x squared. And so the limit as x approaches 0 from below is in fact 4.

Thinking about the limit as x approaches 0 from above, we'll be dealing with the part of the function with rule 4 plus x. And so the limit as x approaches 0 from above is also 4. And so putting these two limits together, since both the limit as x approaches 0 from below and the limit as x approaches 0 from above are equal to the same value, we say that the limit of f of x as x approaches 0 exists and it does in fact equal 4.

We also note that the value of the function at 0 is also 4, so the function is defined at 0 and there is also a limit as x approaches 0. And more importantly than that, since those two things are the same, that is since the limit of f of x as x approaches 0 is equal to the value of the function at 0, we can say that f of x is continuous at x equals 0.

[On an x and y graph, the x axis measures from negative 4 to 4 and the y axis measures from 0 to 10. A curved line marked y equals 4 minus x squared runs up, crossing the x-axis at negative 2 and stopping at (0, 4) on the y-axis. A second line marked y equals 4 plus x starts at (0, 4) on the y-axis and runs diagonally upwards.]

NARRATOR: Consideration of the graph of y equals f of x also allows us to informally determine continuity at x equals 0. Since the graph of y equals f of x can be drawn through x equals 0 without lifting the pen from the paper, we can say that f of x is continuous at x equals 0.