When you work with limit and continuity problems in calculus, there are a couple of formal definitions you need to know about. So, before you take on the following practice problems, you should first re-familiarize yourself with these definitions.

Here is the formal, three-part definition of a limit:

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For a function f (x) and a real number a,

The area of a regular pentagon with side length a 0 a 0 is pa 2 with p = 1 4 5 + 5 + 2 5. P = 1 4 5 + 5 + 2 5. The Pentagon in Washington, DC, has inner sides of length 360 ft and outer sides of length 920 ft. Write an integral to express the area of the roof of the Pentagon according to these dimensions and evaluate this area. In calculus, a function is continuous at x = a if - and only if - all three of the following conditions are met: The function is defined at x = a; that is, f(a) equals a real number The limit of.

exists if and only if

1.4 Continuity Calculus

(Note that this definition does not apply to limits as x approaches infinity or negative infinity.)

Now, here’s the definition of continuity:

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1.4 Continuityap Calculus Definition

A function f (x) is continuous at a point a if three conditions are satisfied:

1.4 Continuityap Calculus Calculator

Now it’s time for some practice problems.

1.4 Continuityap Calculus Test

Practice questions

Using the definitions and this figure, answer the following questions.

1. At which of the following x values are all three requirements for the existence of a limit satisfied, and what is the limit at those x values?

   x = –2, 0, 2, 4, 5, 6, 8, 10, and 11.

2. At which of the x values are all three requirements for continuity satisfied?

Answers and explanations

1. All three requirements for the existence of a limit are satisfied at the x values 0, 4, 8, and 10:

   At 0, the limit is 2.

   At 4, the limit is 5.

   At 8, the limit is 3.

   At 10, the limit is 5.

   To make a long story short, a limit exists at a particular x value of a curve when the curve is heading toward some particular y value and keeps heading toward that y value as you continue to zoom in on the curve at the x value. The curve must head toward that y value (that height) as you move along the curve both from the right and from the left (unless the limit is one where x approaches infinity).

   The phrase heading toward is emphasized here because what happens precisely at the given x value isn’t relevant to this limit inquiry. That’s why there is a limit at a hole like the ones at x = 8 and x = 10.

2. The function in the figure is continuous at 0 and 4.

   The common-sense way of thinking about continuity is that a curve is continuous wherever you can draw the curve without taking your pen off the paper. It should be obvious that that’s true at 0 and 4, but not at any of the other listed x values.