1.6 Limit Based Continuityap Calculus

Limits and continuity concept is one of the most crucial topics in calculus. Combination of these concepts have been widely explained in Class 11 and Class 12. A limit is defined as a number approached by the function as an independent function’s variable approaches a particular value. 1:6 2:56 Nevertheless, if you compute x2 for some values of xbetween 1 and 2, and check if you get more or less than 2, then it looks like there should be some number xbetween 1:4 and 1:5 whose square is exactly 2. So, we assume that there is such a number, and we call it the square root of 2, written as p 2. This raises several questions.

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Continuity and Discontinuity

Functions which have the characteristic that their graphs can bedrawn without lifting the pencil from the paper are somewhat special,in that they have no funny behaviors. The property which describes thischaracteristic is called continuity.

Definition of Continuity at a Point

A function $f(x)$ is continuous at a point where $x=c$ when the following three conditions are satisfied.

  • The function exists at $x=c$. (In other words, $f(c)$ is a real number.)
  • The limit of the function exists at $x=c$. (That is, $limlimits_{xto c}f(x)$ is a real number.)
  • The two values are equal. (That is, $limlimits_{xto c}f(x)=f(c)$.)

If a function has a hole, the three conditions effectively insistthat the hole be filled in with a point to be a continuous function.

More Definitions

Continuity can also be defined on one side of a point, using a one-sided limit.

1.6 Limit Based Continuityap Calculus2nd
  • A function $f(x)$ is continuous from the left at the value $x=c$ when $f(c)$ exists, $limlimits_{xto c-}f(x)$ exists, and $limlimits_{xto c-}f(x)=f(c)$.
  • A function $f(x)$ is continuous from the right at the value $x=c$ when $f(c)$ exists, $limlimits_{xto c+}f(x)$ exists, and $limlimits_{xto c+}f(x)=f(c)$.

We can also define continuity on an interval.

  • A function $f(x)$ is continuous on the open interval $(a,b)$ if it is continuous at every point $x=c$ contained in that interval.
  • A function $f(x)$ is continuous on the closed interval $[a,b]$ if it is continuous on the open interval $(a,b)$, it is continuous from the right at $x=a$, and continuous from the left at $x=b$.
  • A function $f(x)$ is continuous everywhere if it is continuous at every point on the interval $(-infty,infty)$.

Rather than define whether a function is continuous or not, it is more useful to determine where a function is continuous.

Testing for Continuity

Show that the function $f(x)=left{matrix{ dfrac{2x-6}{x-3} & text{when } xne 3 2 & text{when } x=3}right}$ is continuous at $x=3$.

Note that $f(3)=2$, by the definition of $f(x)$.Therefore the function exists, and the first condition is met.
Also, $limlimits_{xto 3}dfrac{2x-6}{x-3}=limlimits_{xto 3}dfrac{2(x-3)}{x-3}=limlimits_{xto 3}2=2$.The limit exists, and the second condition is met.
And this implies $limlimits_{xto 3}dfrac{2x-6}{x-3}=f(3)$. Therefore, $f(x)$ is continuous at $x=3$.The equality demonstrates the third and last condition.

To prove a function is not continuous, it is sufficient to show that one of the three conditions stated above is not met.

1.6 limit based continuityap calculus 14th edition

Types of Discontinuity

When a function is not continuous at a point, then we can say it isdiscontinuous at that point. There are several types of behaviors thatlead to discontinuities.

A removable discontinuity exists when the limit of thefunction exists, but one or both of the other two conditions is notmet. The graphical feature that results is often colloquially called ahole. The first graph below shows a function whose value at $x=c$ is not defined. The second graph below shows a function which has both a limit and a value at $x=c$, but the two values are not equal. This type of function is frequently encountered when trying to find slopes of tangent lines.

An infinite discontinuity exists when one of the one-sided limits of the function is infinite. In other words, $limlimits_{xto c+}f(x)=infty$, or one of the other three varieties of infinite limits. If the twoone-sided limits have the same value, then the two-sided limit willalso exist. Graphically, this situation corresponds to a verticalasymptote. Many rational functions exhibit this type of behavior.

A finite discontinuity exists when the two-sided limit doesnot exist, but the two one-sided limits are both finite, yet not equalto each other. The graph of a function having this feature will show avertical gap between the two branches of the function. The function $f(x)=dfrac{ x }{x}$ has this feature. The graph below is of a generic function with a finite discontinuity.

An oscillating discontinuity exists when the values of thefunction appear to be approaching two or more values simultaneously. Astandard example of this situation is the function $f(x)=sinleft(dfrac{1}{x}right)$, pictured below.

1.6 Limit Based Continuityap Calculus 14th Edition

It is possible to construct functions with even strangerdiscontinuities. Often, mathematicians will refer to these examples as'pathological', because their behavior can seem very counterintuitive.One such example is the function $f(x)=left{matrix{ x & text{when }xtext{ is rational} 2 & text{when }xtext{ is irrational}}right}$. This function can be proven to be continuous at exactly one point only. An approximation of its graph is shown below.

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:

1.6 Limit Based Continuityap Calculus

1.6 Limit Based Continuityap Calculus Algebra

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

exists if and only if

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

Now, here’s the definition of continuity:

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

1.6 Limit Based Continuityap Calculus 2nd Edition

Now it’s time for some practice problems.

Practice questions

1.6 Limit Based Continuityap Calculus Calculator

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. Hp envy 4520 mac os.

    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.