What is the Definition of a Limit for a Function Approaching Negative Infinity?

In summary, the conversation is about defining the statement \lim_{x\rightarrow-\infty}f(x)=L and two possible definitions are discussed. One definition involves the use of limits, while the other involves the use of epsilon. Both definitions do not require N to be an integer.
  • #1
azatkgz
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Homework Statement



Given a function [tex]f:R\rightarrow R[/tex] and a number L,write down a definition of the statement

[tex]\lim_{x\rightarrow-\infty}f(x)=L[/tex]


The Attempt at a Solution



Is it just [tex]\lim_{x\rightarrow-\infty}f(x)=\lim_{x\rightarrow\infty}f(-x)[/tex] ?

and definition is
for [tex]\forall \epsilon>0[/tex] [tex]\exists N[/tex] such that [tex]\forall n>N[/tex]
we have [tex]|f(-x)-L|<\epsilon[/tex]
 
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  • #2
assuming by n you mean x, then yes, this looks like a good dfn, although the usual dfn is that "for all e>0, there is an N<0 such that x<N ==>|f(x)-L|<e"
 
  • #3
Good.Thanks.
 
  • #4
A more "standard" definition of
[tex]\lim_{x\rightarrow-\infty}f(x)=L[/tex]
would be:

"Given [itex]\epsilon> 0[/itex], there exist N such that if x< N, then [itex]|f(x)-L|<\epsilon[/itex]."

Notice that in neither this definition nor your definition is N required to be an integer.
 

Question 1: What is the definition of a limit?

The definition of a limit in mathematics is the value that a function approaches as the input (x-value) approaches a specific value. It is represented by the notation lim f(x) as x approaches a.

Question 2: How can limits be used in calculus?

Limits are used in calculus to help determine the behavior of a function, such as whether it approaches a specific value or becomes infinitely large. They are also used to calculate derivatives and integrals.

Question 3: What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit involves only approaching the specified value from one direction, either from the left or the right. A two-sided limit involves approaching the value from both directions, and the function must approach the same value from both sides for the limit to exist.

Question 4: Can a limit exist if the function is not defined at that point?

Yes, a limit can exist even if the function is not defined at that point. As long as the function approaches the same value from both sides, the limit exists. However, if the function has a discontinuity at that point, the limit does not exist.

Question 5: Are there any special cases where a limit does not exist?

Yes, there are two special cases where a limit does not exist: oscillating behavior, where the function does not approach a specific value but instead oscillates between two values, and approaching infinity, where the function grows without bound as the input approaches a specific value.

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