Calculating limits of function
up vote
1
down vote
favorite
On practical classes at university we calculated some limits of functions of the following type:
$$lim_{xto 0}f(x)^{k(x)}$$
The key idea was that we know that $exists lim_{xto 0}f(x) = p in mathbb{R}$ (usually $p = e$). So we put this result in our initial expression:
$$lim_{xto 0}f(x)^{k(x)} = lim_{xto 0}p^{k(x)}$$
How can this step be proved formally? Probably, to prove this should be used variable substitution, but in such case I have ho ideas how to deal with $k(x)$.
upd: Ok, this cannot be proved in general, but what if $f(x) =(1+x)^{frac{1}{x}}$?
real-analysis limits
asked Nov 26 at 16:01
Yan Kardziyaka
83
add a comment |
up vote
1
down vote
favorite
On practical classes at university we calculated some limits of functions of the following type:
$$lim_{xto 0}f(x)^{k(x)}$$
The key idea was that we know that $exists lim_{xto 0}f(x) = p in mathbb{R}$ (usually $p = e$). So we put this result in our initial expression:
$$lim_{xto 0}f(x)^{k(x)} = lim_{xto 0}p^{k(x)}$$
How can this step be proved formally? Probably, to prove this should be used variable substitution, but in such case I have ho ideas how to deal with $k(x)$.
upd: Ok, this cannot be proved in general, but what if $f(x) =(1+x)^{frac{1}{x}}$?
real-analysis limits
asked Nov 26 at 16:01
Yan Kardziyaka
83
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
On practical classes at university we calculated some limits of functions of the following type:
$$lim_{xto 0}f(x)^{k(x)}$$
The key idea was that we know that $exists lim_{xto 0}f(x) = p in mathbb{R}$ (usually $p = e$). So we put this result in our initial expression:
$$lim_{xto 0}f(x)^{k(x)} = lim_{xto 0}p^{k(x)}$$
How can this step be proved formally? Probably, to prove this should be used variable substitution, but in such case I have ho ideas how to deal with $k(x)$.
upd: Ok, this cannot be proved in general, but what if $f(x) =(1+x)^{frac{1}{x}}$?
real-analysis limits
asked Nov 26 at 16:01
Yan Kardziyaka
83
On practical classes at university we calculated some limits of functions of the following type:
$$lim_{xto 0}f(x)^{k(x)}$$
The key idea was that we know that $exists lim_{xto 0}f(x) = p in mathbb{R}$ (usually $p = e$). So we put this result in our initial expression:
$$lim_{xto 0}f(x)^{k(x)} = lim_{xto 0}p^{k(x)}$$
How can this step be proved formally? Probably, to prove this should be used variable substitution, but in such case I have ho ideas how to deal with $k(x)$.
upd: Ok, this cannot be proved in general, but what if $f(x) =(1+x)^{frac{1}{x}}$?
real-analysis limits
real-analysis limits
asked Nov 26 at 16:01
Yan Kardziyaka
83
asked Nov 26 at 16:01
Yan Kardziyaka
83
edited Nov 26 at 18:23
asked Nov 26 at 16:01
Yan Kardziyaka
83
asked Nov 26 at 16:01
Yan Kardziyaka
83
asked Nov 26 at 16:01
Yan Kardziyaka
83
83
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
This can't be proven formally, because it isn't true in general. Take $f(x) = 1 - x$, and $k(x) = frac{1}{x}$. Then $p = 1$, and so $lim_{xto 0}p^{k(x)} = 1$, but $lim_{xto 0}f(x)^{k(x)} = frac{1}{e}$.
answered Nov 26 at 16:06
user3482749
2,086414
thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
– Yan Kardziyaka
Nov 26 at 16:22
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
This can't be proven formally, because it isn't true in general. Take $f(x) = 1 - x$, and $k(x) = frac{1}{x}$. Then $p = 1$, and so $lim_{xto 0}p^{k(x)} = 1$, but $lim_{xto 0}f(x)^{k(x)} = frac{1}{e}$.
answered Nov 26 at 16:06
user3482749
2,086414
thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
– Yan Kardziyaka
Nov 26 at 16:22
add a comment |
up vote
2
down vote
accepted
This can't be proven formally, because it isn't true in general. Take $f(x) = 1 - x$, and $k(x) = frac{1}{x}$. Then $p = 1$, and so $lim_{xto 0}p^{k(x)} = 1$, but $lim_{xto 0}f(x)^{k(x)} = frac{1}{e}$.
answered Nov 26 at 16:06
user3482749
2,086414
thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
– Yan Kardziyaka
Nov 26 at 16:22
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
This can't be proven formally, because it isn't true in general. Take $f(x) = 1 - x$, and $k(x) = frac{1}{x}$. Then $p = 1$, and so $lim_{xto 0}p^{k(x)} = 1$, but $lim_{xto 0}f(x)^{k(x)} = frac{1}{e}$.
answered Nov 26 at 16:06
user3482749
2,086414
This can't be proven formally, because it isn't true in general. Take $f(x) = 1 - x$, and $k(x) = frac{1}{x}$. Then $p = 1$, and so $lim_{xto 0}p^{k(x)} = 1$, but $lim_{xto 0}f(x)^{k(x)} = frac{1}{e}$.
answered Nov 26 at 16:06
user3482749
2,086414
answered Nov 26 at 16:06
user3482749
2,086414
answered Nov 26 at 16:06
user3482749
2,086414
answered Nov 26 at 16:06
user3482749
2,086414
2,086414
thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
– Yan Kardziyaka
Nov 26 at 16:22
add a comment |
thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
– Yan Kardziyaka
Nov 26 at 16:22
thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
– Yan Kardziyaka
Nov 26 at 16:22
thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
– Yan Kardziyaka
Nov 26 at 16:22
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014500%2fcalculating-limits-of-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
