Limit of compositions at infinity.

In summary, the equality lim_{x\rightarrow a}f(g(x)) = f(lim_{x\rightarrow a} g(x)) holds if f(x) is continuous at g(a). However, if lim_{x\rightarrow a}g(x)=\infty, the equality is not well defined. In this case, it is correct to write lim_{x\rightarrow a}f(g(x))=lim_{x\rightarrow\infty}f(x), but the limit may or may not exist and may be finite or infinite. Continuity is not defined at infinity.
  • #1
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We often have

[tex]lim_{x\rightarrow a}f(g(x)) = f(lim_{x\rightarrow a} g(x))[/tex]

if f(x) is continuous at g(a).

But then my question arises where [tex]g(x)\rightarrow\infty[/tex]. I am not sure if there is any meaning to continuity at infinity as it seems that continuity is the property of a particular point. If the function is proven to be continuous for all x or at least for large x then will this equality hold?
 
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  • #2
That equality holds if f(x) is continuous at [tex]lim_{x\rightarrow a}g(x)[/tex] (which is g(a) if g is continuous at a).

If [tex]lim_{x\rightarrow a}g(x)=\infty[/tex] then the equality isn't well defined, in this case it is correct to write that

[tex]lim_{x\rightarrow a}f(g(x))=lim_{x\rightarrow\infty}f(x)[/tex].

This last limit might exist or not and might be finite or infinite. For example for f(x)=x the limit gives infinite, for f(x)=1/x (f here is continuous in R-{0}) it is 0 and for f(x)=sinx it does not exist.

P.S Continuity is not defined at infinity.
 
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1. What is the concept of limit of compositions at infinity?

The limit of compositions at infinity is a mathematical concept that refers to the behavior of a composition of functions as the input of the functions approaches infinity. It helps in understanding the behavior of a function as its input values increase without bound.

2. How is the limit of compositions at infinity calculated?

The limit of compositions at infinity can be calculated by first finding the limit of each individual function in the composition as the input approaches infinity. Then, the limits are combined using algebraic operations to find the overall limit of the composition.

3. What is the significance of the limit of compositions at infinity in real-world applications?

The limit of compositions at infinity is used in various fields such as physics, engineering, and economics to model real-world phenomena. It helps in predicting the behavior of systems as they approach infinite values, which is often the case in real-life situations.

4. Can the limit of compositions at infinity be infinite?

Yes, the limit of compositions at infinity can be infinite. This means that as the input of the functions in the composition increases without bound, the output of the composition also increases without bound.

5. How is the concept of limit of compositions at infinity related to the concept of limits at infinity?

The limit of compositions at infinity is a special case of the concept of limits at infinity. While the limit of compositions at infinity deals with compositions of functions, the concept of limits at infinity deals with individual functions and their behavior as the input approaches infinity.

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