# Questions tagged [epsilon-delta]

For questions regarding $\varepsilon$-$\delta$ definitions of limits and continuity.

1,670 questions
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### is my epsilon-delta proof correct?

I am a complete beginner with limits and I self study, so I don't have anyone to confirm my answers. I had a simple limit to prove with the precise definition, its a linear equation and I did lots of ...
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### Epsilon Delta proof containing decimal exponents

so I have to prove lim x->infinity ((x^0.8)/(1+x^0.9)) = 0 I am just introduced to epsilon delta, and have no idea how to do this. Please help :( Thanks!
1answer
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### Elementary proof of $\lim\limits_{x\to 0}a^x = 1$ when $a>0$

I have a proof which uses sequences that converge to $0$. I think there is an easier proof than mine but I couldn't find. Is there anyone can prove this without using sequences? Or would you please ...
1answer
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### Can continuous functions have removable discontinuities?

I'm trying to resolve what seems like an inconsistency between the epsilon-delta definition of continuity and the limit-based definition ($\lim_{x->c} f(x) = f(c)$). Assume $c$ is a cluster point. ...
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### Understanding proof of sequential continuity?

I'm trying to understand proof of the following statement: Q. Let $f$ be a function on a closed bounded interval $[a,b]$. Prove that $f$ is continuous at $c \in [a,b]$ if and only if $f(x_n) \to c$...
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### How do I prove using an $\epsilon - \delta$ proof that $\lim_{x\rightarrow \frac{1}{e}}(e^{x^{x^x}})<2$?

Not a homework question. Just wanting to refresh my epsilon delta proofs, and came up with this - struggled for an hour, no idea where to start.
3answers
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### Using the $\varepsilon − N$ definition of the limit, prove that $\lim \limits_{n\to\infty} \frac{(n^2 + 1)}{ (n^2 + 2)} = 1$.

Using the $ε − N$ definition of the limit, prove that $\displaystyle\lim \limits_{n\to\infty} \frac{(n^2 + 1)}{ (n^2 + 2)} = 1$. In other words, given $\varepsilon> 0$, find explicitly a natural ...
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### Question about the proof of the $\epsilon-\delta$ definition of continuity

I am currently trying to get my head around the proof of the definition of continuity of a function given in my Elementary Analysis textbook. The definition given is: Let $f$ be a real-valued function ...
1answer
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### Intuitive explanation of sign preservation of limits and boundedness?

Is there an intuitive explanation of the following two statements? Let $I \subseteq \mathbb{R}$ be an open interval, let $c \in I$ and let $f:I - \left\{c\right\} \to \mathbb{R}$ be a function. If ...
1answer
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### proof lim x-a f(x)= lim x-0 f(x+a) ( duplicate)

i guess the proof here(Formal proof of $\lim_{x\to a}f(x) = \lim_{h\to 0} f(a+h)$) is something wrong, its too easy and i thought i couldnt change things like the one most voted
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### Whether the product of uniformly continuous functions is uniformly continuous [closed]

I know it isn't and I have to give a counter-example. Function $f_1(x)=f_2(x)=x$ this is a uniformly continuous function the product of these functions $f_1(x)\cdot f_2(x)=x\cdot x=x^2$ this isn't an ...