InfiniteLimits

__Infinite Limits__ Created by: Evan Sweet, Becca LaBo and Porsha Waller

Websites and links:
[|Limits to Infinity] **[|Cliff Notes-infinite limits]** [|Infinite Limits-Video] [|Limits (Introduction)] [|Limits and Infinity]

In this section we will take a look at limits whose value is infinity or minus infinity. These kinds of limit will show up fairly regularly in later sections and in other courses and so you’ll need to be able to deal with them when you run across them.

The first thing we should probably do here is to define just what we mean when we sat that a limit has a value of infinity or minus infinity. __**Definition**__ We Say **lim f(x)=infinity** **x->a** if we can make f(x) arbitrarily large for all //x// sufficiently close to //x=a//, from both sides, without actually letting x=a.

We Say **lim f(x)=negative infinity** **x->a** if we can make f(x) arbitrarily large and negative for all //x//sufficiently close to x=a, from both sides, without actually letting x=a.

These definitions can be appropriately modified for the one-sided limits as well. To see a more precise and mathematical definition of this kind of limit see the [|The Definition of the Limit] section at the end of this chapter.



We say that a variable "becomes infinite" if, beginning with a certain term of a [|sequence] of its values, the absolute value of that term and any subsequent term we might name is greater than any positive number we name, however large. When the variable is //x//, and it takes on only positive values, then it becomes positively infinite. We write x->infinity If //x// takes on only negative values, it becomes negatively infinite, in which case we write x-> negative infinity In both cases, we mean: No matter what large number M we might name, we get to a point in the sequence of values of //x// that their absolute values become greater than M. When the variable is f(x) and it becomes positively or negatively infinite when //x// approaches the value //c//, then we write Lim f(x) = infinity Lim f(x) = negative infinity x->c x->c Although we write the symbol "lim" for limit, those algebraic statements mean: The limit of f(x) as //x// approaches //c// does not exist. For a limit is a number And when a variable becomes infinite, its values increase without bound. Definition of "becomes infinite;" it is not the definition of a [|limit]. Thus we employ the symbol infinity in algebraic statements to signify that the definition of [|becomes infinite] has been satisfied. That symbol by itself has no meaning. //x// || : || 
 * As an example, here is the graph of the function || //y// || = || __1__



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ASSESSMENT

[|Infinite Limits Assessment]