Prove that $lim_a to 1 a^1/n = 1$ using õ-ô definition
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Question as above. I have factorised $a-1$ into $(a^frac1n-1)(1+a^-1+...+a^-n)$ and stucked all the way. How could I continue the way to prove using delta-epsilon definition of limit?
analysis
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up vote
1
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Question as above. I have factorised $a-1$ into $(a^frac1n-1)(1+a^-1+...+a^-n)$ and stucked all the way. How could I continue the way to prove using delta-epsilon definition of limit?
analysis
The factorisation is wrong.
â user90369
Aug 28 at 14:51
What's $n$ here dear ?
â Nosrati
Aug 28 at 14:54
$n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
â Wei Lam
Aug 28 at 15:19
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Question as above. I have factorised $a-1$ into $(a^frac1n-1)(1+a^-1+...+a^-n)$ and stucked all the way. How could I continue the way to prove using delta-epsilon definition of limit?
analysis
Question as above. I have factorised $a-1$ into $(a^frac1n-1)(1+a^-1+...+a^-n)$ and stucked all the way. How could I continue the way to prove using delta-epsilon definition of limit?
analysis
asked Aug 28 at 14:49
Wei Lam
83
83
The factorisation is wrong.
â user90369
Aug 28 at 14:51
What's $n$ here dear ?
â Nosrati
Aug 28 at 14:54
$n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
â Wei Lam
Aug 28 at 15:19
add a comment |Â
The factorisation is wrong.
â user90369
Aug 28 at 14:51
What's $n$ here dear ?
â Nosrati
Aug 28 at 14:54
$n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
â Wei Lam
Aug 28 at 15:19
The factorisation is wrong.
â user90369
Aug 28 at 14:51
The factorisation is wrong.
â user90369
Aug 28 at 14:51
What's $n$ here dear ?
â Nosrati
Aug 28 at 14:54
What's $n$ here dear ?
â Nosrati
Aug 28 at 14:54
$n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
â Wei Lam
Aug 28 at 15:19
$n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
â Wei Lam
Aug 28 at 15:19
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Hint:
I'm assuming that $ninmathbbN$.
As suggested in the comments, you made a mistake when computing the factorization. After you find the correct one, you will see that
$$ leftvert a^frac1n-1 rightvert = fracvert a-1vertleftvert a^fracn-1n+ldots+a^frac1n+1 rightvert leq vert a-1vert, $$
given that $ageq 0$.
Can you finish?
@MarkViola Oh, indeed. I will add that to my answer, thanks.
â Sobi
Aug 28 at 15:04
2
Take $ô=õ$ to complete the proof?
â Wei Lam
Aug 28 at 15:11
@WeiLam Yes! $$
â Sobi
Aug 28 at 15:14
1
@Sobi Thanks all the way!! I never thought this question could stuck me for few days due to my incorrect way of factorisation. I get doubted when I was trying to prove it using limit definition.
â Wei Lam
Aug 28 at 15:17
@WeiLam No need to thank me, you basically did all the work yourself!
â Sobi
Aug 28 at 15:19
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Hint:
I'm assuming that $ninmathbbN$.
As suggested in the comments, you made a mistake when computing the factorization. After you find the correct one, you will see that
$$ leftvert a^frac1n-1 rightvert = fracvert a-1vertleftvert a^fracn-1n+ldots+a^frac1n+1 rightvert leq vert a-1vert, $$
given that $ageq 0$.
Can you finish?
@MarkViola Oh, indeed. I will add that to my answer, thanks.
â Sobi
Aug 28 at 15:04
2
Take $ô=õ$ to complete the proof?
â Wei Lam
Aug 28 at 15:11
@WeiLam Yes! $$
â Sobi
Aug 28 at 15:14
1
@Sobi Thanks all the way!! I never thought this question could stuck me for few days due to my incorrect way of factorisation. I get doubted when I was trying to prove it using limit definition.
â Wei Lam
Aug 28 at 15:17
@WeiLam No need to thank me, you basically did all the work yourself!
â Sobi
Aug 28 at 15:19
add a comment |Â
up vote
1
down vote
accepted
Hint:
I'm assuming that $ninmathbbN$.
As suggested in the comments, you made a mistake when computing the factorization. After you find the correct one, you will see that
$$ leftvert a^frac1n-1 rightvert = fracvert a-1vertleftvert a^fracn-1n+ldots+a^frac1n+1 rightvert leq vert a-1vert, $$
given that $ageq 0$.
Can you finish?
@MarkViola Oh, indeed. I will add that to my answer, thanks.
â Sobi
Aug 28 at 15:04
2
Take $ô=õ$ to complete the proof?
â Wei Lam
Aug 28 at 15:11
@WeiLam Yes! $$
â Sobi
Aug 28 at 15:14
1
@Sobi Thanks all the way!! I never thought this question could stuck me for few days due to my incorrect way of factorisation. I get doubted when I was trying to prove it using limit definition.
â Wei Lam
Aug 28 at 15:17
@WeiLam No need to thank me, you basically did all the work yourself!
â Sobi
Aug 28 at 15:19
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Hint:
I'm assuming that $ninmathbbN$.
As suggested in the comments, you made a mistake when computing the factorization. After you find the correct one, you will see that
$$ leftvert a^frac1n-1 rightvert = fracvert a-1vertleftvert a^fracn-1n+ldots+a^frac1n+1 rightvert leq vert a-1vert, $$
given that $ageq 0$.
Can you finish?
Hint:
I'm assuming that $ninmathbbN$.
As suggested in the comments, you made a mistake when computing the factorization. After you find the correct one, you will see that
$$ leftvert a^frac1n-1 rightvert = fracvert a-1vertleftvert a^fracn-1n+ldots+a^frac1n+1 rightvert leq vert a-1vert, $$
given that $ageq 0$.
Can you finish?
edited Aug 28 at 15:04
answered Aug 28 at 15:01
Sobi
2,623415
2,623415
@MarkViola Oh, indeed. I will add that to my answer, thanks.
â Sobi
Aug 28 at 15:04
2
Take $ô=õ$ to complete the proof?
â Wei Lam
Aug 28 at 15:11
@WeiLam Yes! $$
â Sobi
Aug 28 at 15:14
1
@Sobi Thanks all the way!! I never thought this question could stuck me for few days due to my incorrect way of factorisation. I get doubted when I was trying to prove it using limit definition.
â Wei Lam
Aug 28 at 15:17
@WeiLam No need to thank me, you basically did all the work yourself!
â Sobi
Aug 28 at 15:19
add a comment |Â
@MarkViola Oh, indeed. I will add that to my answer, thanks.
â Sobi
Aug 28 at 15:04
2
Take $ô=õ$ to complete the proof?
â Wei Lam
Aug 28 at 15:11
@WeiLam Yes! $$
â Sobi
Aug 28 at 15:14
1
@Sobi Thanks all the way!! I never thought this question could stuck me for few days due to my incorrect way of factorisation. I get doubted when I was trying to prove it using limit definition.
â Wei Lam
Aug 28 at 15:17
@WeiLam No need to thank me, you basically did all the work yourself!
â Sobi
Aug 28 at 15:19
@MarkViola Oh, indeed. I will add that to my answer, thanks.
â Sobi
Aug 28 at 15:04
@MarkViola Oh, indeed. I will add that to my answer, thanks.
â Sobi
Aug 28 at 15:04
2
2
Take $ô=õ$ to complete the proof?
â Wei Lam
Aug 28 at 15:11
Take $ô=õ$ to complete the proof?
â Wei Lam
Aug 28 at 15:11
@WeiLam Yes! $$
â Sobi
Aug 28 at 15:14
@WeiLam Yes! $$
â Sobi
Aug 28 at 15:14
1
1
@Sobi Thanks all the way!! I never thought this question could stuck me for few days due to my incorrect way of factorisation. I get doubted when I was trying to prove it using limit definition.
â Wei Lam
Aug 28 at 15:17
@Sobi Thanks all the way!! I never thought this question could stuck me for few days due to my incorrect way of factorisation. I get doubted when I was trying to prove it using limit definition.
â Wei Lam
Aug 28 at 15:17
@WeiLam No need to thank me, you basically did all the work yourself!
â Sobi
Aug 28 at 15:19
@WeiLam No need to thank me, you basically did all the work yourself!
â Sobi
Aug 28 at 15:19
add a comment |Â
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The factorisation is wrong.
â user90369
Aug 28 at 14:51
What's $n$ here dear ?
â Nosrati
Aug 28 at 14:54
$n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
â Wei Lam
Aug 28 at 15:19