# Tag Info

## New answers tagged epsilon-delta

### $\epsilon$-$\delta$ proof, finding $\delta$

Could we have have used the inequality: for all positive $x$ less than $1$, $$|\log(x)| \leq \dfrac{1}{\sqrt{x}}.$$ I guessed it from the fact that $$\lim\limits_{x\to 0^{+}} \sqrt{x}\log(x)=0.$$ Then,...
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### $\epsilon$-$\delta$ proof, finding $\delta$

I think you can easily, changing $x$ by $\lfloor x\rfloor$, reduce the asked (write, if not) case to a fraction $\frac{\log _{2}n}{n}, n\in \mathbb{N}$ and let me show some technique for its ...
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### Epsilon proof verification

The calculation is all fine, but it falls short at the end. Setting $\varepsilon$ to be a function of $K$ is a misstep; you don't get to determine what form $\varepsilon$ takes, as it is an arbitrary ...
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### Show that $\lim_{x\to 4} \sqrt{x} = 2$ using $\epsilon$-$\delta$ definition of limit.

As noted in the comments, both your methods are wrong due to a wrong choice of $\delta$. One thing I find helpful when trying to proof these kind of limits is to choose multiple $\delta$ for different ...
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### Proof that f is not continuous

The definition of limit is that for any $\epsilon > 0$, there exists $\delta > 0$ such that whenever $0 < |x-a| < \delta$, then $|f(x) - L| < \epsilon$. The idea is that if there ...
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### Why is the condition $0<|x-a|<\delta$ necessary in the $\varepsilon$-$\delta$ definition of a limit?

EDIT: In response to your edit, yes, your last explanation is right--the concept of "closeness" is the whole point of the limit definition. If $x$ is "close" to $a$, then $f(x)$ is ...
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### Basic analysis question to estimate the growth of a function

Let's say you got the result for $t \in (-\delta, \delta) \cup (-\infty, A) \cup (B, \infty)$ with for example $A \leq -\delta$, $\delta \leq B$ to make the next expression easy to write. Now, the ...
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