When is it OK to pass the limit to the exponent

In summary, the conversation discusses the justification for performing a limit operation with exponential functions and a composite function involving a limit. The concept of continuity is mentioned, with the conclusion that since exponential functions are continuous everywhere, the limit operation can be performed in both cases.
  • #1
clueless...
2
0
when is it justified to do something like this

[tex] \lim_{n\to\infty} e^{lnx^{1/n}} = e^{\displaystyle\lim_{n\to\infty}lnx^{1/n}}[/tex]

or something like this

[tex] \lim_{n\to\infty} 2^{f(n)} = 2^{\displaystyle\lim_{n\to\infty}f(n)}}[/tex] ?

I am assuming that I can do something like this in both cases, but why?

thank you.
 
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  • #2
in general
lim f(g(n))=f(lim g(n))
holds when f is continuous at lim g(n)
exponential functions are everywhere continuous so this can be done in both cases
 
  • #3
In fact [itex]\lim_{x\to a}f(x)= f(\lim_{x\to a} x)= f(a)[/itex], from which [itex]\lim_{x\to a}f(g(x))= f(\lim_{x\to a}g(x))[/itex]
is the definition of "f is continuous at x= a".
 
  • #4
Wherever it is continuous which is everywhere
 

1. When is it OK to pass the limit to the exponent?

It is generally considered acceptable to pass the limit to the exponent in cases where the limit exists and is finite. In other words, if the limit approaches a constant value as the exponent increases without bound, then it is acceptable to pass the limit to the exponent.

2. How do I know if the limit exists?

To determine if the limit exists, you must evaluate the function at the limit point from both the left and right sides. If the values are equal, then the limit exists. If the values are different or approach different values, then the limit does not exist.

3. Can I always pass the limit to the exponent?

No, there are cases where passing the limit to the exponent is not valid. For example, if the limit is approaching infinity or negative infinity, then passing the limit to the exponent would result in an undefined or indeterminate value.

4. Is there a specific rule for passing the limit to the exponent?

Yes, there is a rule known as the "limit laws" that states that the limit of a function raised to a power is equal to the limit of the function raised to that same power. This rule can be applied in most cases when passing the limit to the exponent.

5. Can passing the limit to the exponent change the result?

Yes, passing the limit to the exponent can sometimes change the result. This is because the limit itself may change as the exponent increases. Therefore, it is important to carefully evaluate the limit before passing it to the exponent to ensure accurate results.

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