Calculus


 * || || Introductory Calculus: Limit of a Function and Continuity ||
 * || [[image:http://www.algebralab.org/img/f17a4cb4-b048-490f-b1ac-b03d26e7693a.gif width="30" height="30" align="absMiddle"]] **The limit of a [|function] from left or right or both:**
 * || [[image:http://www.algebralab.org/img/f17a4cb4-b048-490f-b1ac-b03d26e7693a.gif width="30" height="30" align="absMiddle"]] **The limit of a [|function] from left or right or both:**

>>>
 * If [[image:http://www.algebralab.org/img/8b2dc7c3-f3f4-44e1-9060-eb3e37d0a9b9.gif]], we call the limit of this [|function] as //x// approaches 1 from the left as

Similarly, we call the limit as //x// approaches 1 from the right equal to 2.

>>> >> >> What is mysterious about this concept of limit ?

The [|function] obviously has a value of 2 when //x// = 1.

>>> We say that

>>>

This is a simple thing to analyze because this [|function] is continuous (the [|graph] is “connected”) and we can clearly see that the [|function] value at x = 1 is 2. See the [|graph] below.




 * Suppose that

>>> >> >> What is the limit as //x// approaches –1 from the left/right? >> >> We will get two different values for this limit because there are two different expressions for. >>> >>> If //x// approaches –1 __from the left__, we use the formula >>> >>>>   >>>> >>> which gives us >>> >>>>   >>> >>> If //x// approaches –1 __from the right__, we use the formula >>> >>>> >>>> >>> which gives us >>> >>>> >> >> In this case, we say that >> >>> >> >> **Since these limits are different, we say that the ONE limit as //x// approaches –1 does not exist. In order for this limit to exist, both the left hand and right hand limits would have to be the same, and the [|graph] would have to “connect” from the left and right sides.** >> >>

>> These limits from the left and right have different values. Looking at a [|graph] from a calculator screen, we can see that the left hand [|graph] and the right hand [|graph] do not meet in one point, but the limits from the left and right sides can be seen on the [|graph] as the //y//values of this [|function] for each piecewise-defined part of the graph. Notice the “open” [|circle] and "closed" [|circle] on the graph. >> >> Notice that the value of the __ [|function] __ at the [|point] –1 is  because only defines this [|function] for the value //x// = –1. This is an important fact as we examine the continuity of a function. We will compare this value, if it exists, to the limit value. >>   **Continuity of a [|function] at a [|point] x = a:** Definition || A [|function] is said to be continuous at a [|point] where x = a if three conditions hold: >>> ||

>> >>> >>> >> we apply the definition of continuity at the [|point] x = –1. >>> >>> Condition (i) is OK because
 * Using our [|function]

>>>> meaning that exists. >>> >>> Condition (ii) is not OK because

>>>> meaning does not exist. >>> >>> We can STOP here because as soon as one of these three conditions goes wrong, we know the [|function] is NOT continuous at the given value of //x//.

>> Find the following limits if they exist.


 * ||  || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] [[image:http://www.algebralab.org/images/exercises.jpg align="absmiddle" caption="Examples"]]Examples: || [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/img/b6d9f8be-83f4-441c-b7b1-e5bcf5a10f5a.gif width="75" height="29"]] ||
 * ||  || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] [[image:http://www.algebralab.org/images/exercises.jpg align="absmiddle" caption="Examples"]]Examples: || [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/img/b6d9f8be-83f4-441c-b7b1-e5bcf5a10f5a.gif width="75" height="29"]] ||


 * [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/img/5eda2a0a-d51f-4c5c-bc86-5db2b6cfee8f.gif width="68" height="44"]] ||


 * [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/img/46a3c82d-b46e-48b0-b6a2-4dcc0290b062.gif width="68" height="44"]] ||


 * [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/img/c9f1dcd5-b571-47d0-9756-60caed7109ad.gif width="68" height="44"]] ||


 * > Use the following two functions to answer the next set of questions.

>> and  ||
 * ||  || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] [[image:http://www.algebralab.org/images/exercises.jpg align="absmiddle" caption="Examples"]]Examples: || [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/img/e6470f5d-56a9-4cf1-ab5e-5a602272030e.gif width="65" height="29"]] ||


 * [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/img/90413e83-2b88-446b-b1a2-44dd19fcb559.gif width="65" height="29"]] ||


 * [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/img/308d6564-850c-4123-968a-d56842c6f3cb.gif width="64" height="29"]] ||


 * [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/img/490fbf3b-223c-416b-828e-6678c39968f3.gif width="61" height="29"]] ||


 * [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/img/435d3af7-eb1e-428f-a592-03074c064f6f.gif width="59" height="29"]]  ||


 * [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/img/0f407651-fb79-4213-b9c7-6c347a87c2ce.gif width="59" height="29"]] ||


 * [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || Discuss the continuity of f(x) and g(x) at x = 2. ||


 * > Test your understanding about limits by answering the following two questions about two new functions: h(x) and r(x). ||
 * ||  || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] [[image:http://www.algebralab.org/images/exercises.jpg align="absmiddle" caption="Examples"]]Examples: || [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || Let [[image:http://www.algebralab.org/img/5d310cd0-83e1-435c-ab27-015496e33c5a.gif width="88" height="44"]]

[|Graph] and discuss continuity at x = 3.

> ||


 * [[image:http://www.algebralab.org/images/spacer35.gif]] || [[image:http://www.algebralab.org/images/spacer35.gif]][[image:http://www.algebralab.org/images/spacer35.gif]] || Let [[image:http://www.algebralab.org/img/b7457ac6-630c-4e63-bb0e-5a87f0447bf0.gif width="149" height="69"]]

[|Graph] and discuss continuity at x = 3.

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 * M Ransom ||
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 * M Ransom ||
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