Linked Questions

4
votes
3answers
525 views

Doubt in $\varepsilon$-$\delta$ proof of continuity of $x^2$ [duplicate]

I have one elementary doubt on the proof that $f(x)=x^2$ is continuous for every $a \in \Bbb R$. The initial steps usually presented are: to deduce which $\delta$ will work we write: $$|x^2-a^2|=|x-a|...
45
votes
5answers
27k views

$\epsilon$-$\delta$ proof that $\lim\limits_{x \to 1} \frac{1}{x} = 1$.

I'm starting Spivak's Calculus and finally decided to learn how to write epsilon-delta proofs. I have been working on chapter 5, number 3(ii). The problem, in essence, asks to prove that $$\lim\...
14
votes
4answers
627 views

How do people pick $\delta$ so fast in $\epsilon$-$\delta$ proofs

For example, in a proof that shows $f(x) = \sqrt x$ is uniformly continuous on the positive real line, the proof goes like: Let $\epsilon > 0$ be given, and $\delta = \epsilon^2$.... Or to show ...
7
votes
4answers
7k views

Prove that $\lim_{x\to 2}x^2=4$ using $\epsilon-\delta$ definition.

Prove that $\lim_{x\to 2}x^2=4$ using $\epsilon-\delta$ definition. By the mean of $\epsilon-\delta$ definition, $|x-2|\le \delta,|x+2|\le \delta+4$ then $|x-2||x+2|\le \delta(\delta+4),|x^2-4|\le \...
6
votes
1answer
20k views

Show Continuity Using Epsilon Delta Definition [closed]

Using Epsilon Delta Definition show that $f(x)=x^2$ is continuous on all R, i.e. that $$\lim_{x\to a} f(x)=f(a)$$ for each $a$ that is an element of the reals. b) do the same for $$ g(x)=\...
11
votes
3answers
7k views

$\epsilon$-$\delta$ proof that $\lim_{x \to 1} \sqrt{x} = 1$

I'm trying to teach myself how to do $\epsilon$-$\delta$ proofs and would like to know if I solved this proof correctly. The answer given (Spivak, but in the solutions book) was very different. ...
2
votes
3answers
5k views

Uniform Continuity of $f(x)=x^3$

1.)Determine whether $f(x)=x^3$ is uniformly continuous on [0,2) So far, I have $\delta$ = 2 and $\epsilon$ = 8, and plan on using the sandwich theorem with $x^2$ and eventually equating $\delta = \...
5
votes
2answers
2k views

I need a better explanation of $(\epsilon,\delta)$-definition of limit

I am reading the $\epsilon$-$\delta$ definition of a limit here on Wikipedia. It says that $f(x)$ can be made as close as desired to $L$ by making the independent variable $x$ close enough, but ...
-2
votes
4answers
997 views

Solving a problem using the definition of limit [closed]

How can I solve this using the definition of limit? Prove using the definition of limit that: $$\lim_{x\to 1} (x²-4x)=-3$$ How can I approach this? EDIT: OH my god! Thanks @adam! Maybe you can ...
0
votes
1answer
4k views

Find limit from first principle

Find the limit or prove the limit does not exist using the definition of the limit: $$\lim\limits_{x \rightarrow c} x^2+x+1.$$ I am getting stuck in the problem following through on the algebra to ...
0
votes
2answers
2k views

Epsilon-Delta questions

Prove the following limits using only the epsilon delta definition: Q1: $$\lim_{x\to2^-} \sqrt{4-x^2}= 0$$ and Q2: $$\lim_{x\to\infty}\dfrac{x^2+2x}{x^2+1} = 1$$ For 1, I got stuck at the ...
2
votes
1answer
2k views

Use epsilon-delta definition to show $f(x)=x^2$ is continuous [duplicate]

Let $f$ be a function $f:\mathbb{R}\to \mathbb{R}$ and $f(x)=x^2.$ Prove that $f$ is continuous on all of $\mathbb{R}$. I tried this: Let $\epsilon >0$ and assume $\mid x-x_0\mid < \delta$ for ...
2
votes
1answer
1k views

Proving $\lim_{x\to 3} (x^2-5x+1)=-5$ by the $\epsilon -\delta$ definition of a limit.

Prove that $\displaystyle\lim_{x\to 3} (x^2-5x+1)=-5$ by the $\epsilon -\delta$ definition of a limit. What I've done so far: $\forall \epsilon >0 \exists \delta\ni 0<|x-3|<\delta\...
1
vote
1answer
948 views

$\epsilon - \delta$ definition to prove that f is a continuous function.

Use the $\epsilon$-$\delta$ definition of continuity to prove that $$f(x)= \frac{x+1}{x-4}$$ is continuous at every point $c \in R \backslash \{4\}$. My attempt is let $c \in R \backslash \{4\}$. ...
2
votes
2answers
531 views

How to complete this epsilon delta proof

Prove $\lim_{x\to 1} {2+4x \over 3} = 2$ using the epsilon delta definition of a limit. if $0 < \left|x-1\right| < \delta$ then $\left|{2+4x \over 3}-2\right| < \epsilon$ scratch work for ...

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