Limit of monotic transformation

In summary, the limit of a monotonic transformation of a function is equal to the monotonic transformation of its limit, as long as the function is continuous at the limit point. The variable n may be replaced with x, and the function f must be continuous at g(a).
  • #1
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Hello, I was just wondering if is it true that the limit of a monotonic transformation of a function is the same as the monotonic transformation of its limit? That is, does

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

for monotonic [tex]f[/tex], some [tex]a[/tex], and such that if the limit does not exist for one side of the expression, it doesn't exist for both?

Thanks.
 
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  • #2
Your original equation does not describe the dependence on n of anything.
 
  • #3
Sorry, replace n with x - was very tired!
 
  • #4
Correction: [tex]\lim_{x \rightarrow a} f(g(x)) = f(\lim_{x \rightarrow a} g(x))[/tex]
 
  • #5
I believe you need f to be continuous at g(a). The g(x) may be a red herring. Just look at lim(x->a)f(x)=f(a).
 

1. What is a limit of monotonic transformation?

A limit of monotonic transformation refers to the maximum or minimum value that a function approaches as its input variable approaches a certain value. In other words, it is the value that the function "approaches" as the input variable gets closer and closer to a specific value.

2. How is the limit of monotonic transformation different from a regular limit?

The limit of monotonic transformation is different from a regular limit in that it only applies to strictly increasing or decreasing functions. Regular limits can apply to any type of function.

3. How is the limit of monotonic transformation used in mathematics?

The limit of monotonic transformation is used in mathematics to analyze the behavior of a function and determine its maximum or minimum value. It is also used in calculus to calculate derivatives and integrals of monotonic functions.

4. What are some examples of monotonic functions?

Some examples of monotonic functions include linear functions, exponential functions, and logarithmic functions. These functions all have a consistent increase or decrease in their values as the input variable changes.

5. How can we determine the limit of monotonic transformation?

The limit of monotonic transformation can be determined by evaluating the function at various points near the input variable in question and observing the trend of the function's values. It can also be calculated using mathematical techniques such as the squeeze theorem or L'Hôpital's rule.

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