Prove that $lim_a to 1 a^1/n = 1$ using ε-δ 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?







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  • The factorisation is wrong.
    – user90369
    Aug 28 at 14:51










  • What's $n$ here dear ?
    – Nosrati
    Aug 28 at 14:54










  • $n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
    – Wei Lam
    Aug 28 at 15:19














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?







share|cite|improve this question




















  • The factorisation is wrong.
    – user90369
    Aug 28 at 14:51










  • What's $n$ here dear ?
    – Nosrati
    Aug 28 at 14:54










  • $n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
    – Wei Lam
    Aug 28 at 15:19












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?







share|cite|improve this question












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?









share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 28 at 14:49









Wei Lam

83




83











  • The factorisation is wrong.
    – user90369
    Aug 28 at 14:51










  • What's $n$ here dear ?
    – Nosrati
    Aug 28 at 14:54










  • $n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
    – Wei Lam
    Aug 28 at 15:19
















  • The factorisation is wrong.
    – user90369
    Aug 28 at 14:51










  • What's $n$ here dear ?
    – Nosrati
    Aug 28 at 14:54










  • $n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
    – Wei Lam
    Aug 28 at 15:19















The factorisation is wrong.
– user90369
Aug 28 at 14:51




The factorisation is wrong.
– user90369
Aug 28 at 14:51












What's $n$ here dear ?
– Nosrati
Aug 28 at 14:54




What's $n$ here dear ?
– Nosrati
Aug 28 at 14:54












$n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
– Wei Lam
Aug 28 at 15:19




$n$ in here stands for natural numbers exclude 0. Feel sorry not to mention it.
– Wei Lam
Aug 28 at 15:19










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?






share|cite|improve this answer






















  • @MarkViola Oh, indeed. I will add that to my answer, thanks.
    – Sobi
    Aug 28 at 15:04







  • 2




    Take $δ=ε$ to complete the proof?
    – Wei Lam
    Aug 28 at 15:11










  • @WeiLam Yes! $$
    – 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.
    – Wei Lam
    Aug 28 at 15:17










  • @WeiLam No need to thank me, you basically did all the work yourself!
    – Sobi
    Aug 28 at 15:19










Your Answer




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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?






share|cite|improve this answer






















  • @MarkViola Oh, indeed. I will add that to my answer, thanks.
    – Sobi
    Aug 28 at 15:04







  • 2




    Take $δ=ε$ to complete the proof?
    – Wei Lam
    Aug 28 at 15:11










  • @WeiLam Yes! $$
    – 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.
    – Wei Lam
    Aug 28 at 15:17










  • @WeiLam No need to thank me, you basically did all the work yourself!
    – Sobi
    Aug 28 at 15:19














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?






share|cite|improve this answer






















  • @MarkViola Oh, indeed. I will add that to my answer, thanks.
    – Sobi
    Aug 28 at 15:04







  • 2




    Take $δ=ε$ to complete the proof?
    – Wei Lam
    Aug 28 at 15:11










  • @WeiLam Yes! $$
    – 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.
    – Wei Lam
    Aug 28 at 15:17










  • @WeiLam No need to thank me, you basically did all the work yourself!
    – Sobi
    Aug 28 at 15:19












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?






share|cite|improve this answer














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?







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








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.
    – Sobi
    Aug 28 at 15:04







  • 2




    Take $δ=ε$ to complete the proof?
    – Wei Lam
    Aug 28 at 15:11










  • @WeiLam Yes! $$
    – 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.
    – Wei Lam
    Aug 28 at 15:17










  • @WeiLam No need to thank me, you basically did all the work yourself!
    – Sobi
    Aug 28 at 15:19
















  • @MarkViola Oh, indeed. I will add that to my answer, thanks.
    – Sobi
    Aug 28 at 15:04







  • 2




    Take $δ=ε$ to complete the proof?
    – Wei Lam
    Aug 28 at 15:11










  • @WeiLam Yes! $$
    – 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.
    – Wei Lam
    Aug 28 at 15:17










  • @WeiLam No need to thank me, you basically did all the work yourself!
    – Sobi
    Aug 28 at 15:19















@MarkViola Oh, indeed. I will add that to my answer, thanks.
– Sobi
Aug 28 at 15:04





@MarkViola Oh, indeed. I will add that to my answer, thanks.
– Sobi
Aug 28 at 15:04





2




2




Take $δ=ε$ to complete the proof?
– Wei Lam
Aug 28 at 15:11




Take $δ=ε$ to complete the proof?
– Wei Lam
Aug 28 at 15:11












@WeiLam Yes! $$
– Sobi
Aug 28 at 15:14




@WeiLam Yes! $$
– 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.
– 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.
– Wei Lam
Aug 28 at 15:17












@WeiLam No need to thank me, you basically did all the work yourself!
– Sobi
Aug 28 at 15:19




@WeiLam No need to thank me, you basically did all the work yourself!
– Sobi
Aug 28 at 15:19

















 

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