Complete the definition: ${\lim }_{x\to a}f(x)=L$ means for every $\epsilon >0$ there is a $\delta >0$ such that ___.

In the epsilon-delta definition of ${\lim }_{x\to a}f(x)=L$, what is the correct order of the two choices?

Why does the definition use $0<|x-a|<\delta$ instead of just $|x-a|<\delta$?

In the definition of ${\lim }_{x\to a}f(x)=L$, which distance must become less than $\epsilon$?

A proof chooses $\delta =\frac{\epsilon }{5}$ after $\epsilon >0$ is given. Why can this match the definition?

For $f(x)=2x-7$, $a=4$, $L=1$ and $\epsilon =0.5$, what is the largest $\delta$ that satisfies the definition?

For $f(x)=5x+1$ at $a=-1$ with $L=-4$ and $\epsilon =0.2$, how large can $\delta$ be at most?

To prove ${\lim }_{x\to 2}(-3x+10)=4$ with $\epsilon =0.9$, what is the largest valid $\delta$?

Given $f(x)=\frac{1}{4}x-2$, $a=8$, $L=0$ and $\epsilon =0.5$, what is the largest $\delta$?

For the limit ${\lim }_{x\to 0}(-6x-1)=-1$ and $\epsilon =1.2$, what exact largest $\delta$ works?

When $f(x)=0.8x+3$, $a=5$, $L=7$ and $\epsilon =0.16$, how far may $x$ be from $a$ at most?

For $f(x)=7x-2$ and any $\epsilon >0$, what is the largest $\delta$ in terms of $\epsilon$ that works?

For $f(x)=-\frac{1}{2}x+4$, what largest $\delta$ in terms of $\epsilon$ works for any center $a$?

For $f(x)=-9x+1$ and any $\epsilon >0$, what is the largest formula for $\delta$ that works?

For $f(x)=0.1x-6$, what largest $\delta$ in terms of $\epsilon$ works for the epsilon-delta definition?

For $f(x)=4x+1$, $a=2$, $L=9$ and $\epsilon =0.12$, does $\delta =0.02$ satisfy the definition?

For $f(x)=-5x+3$, $a=1$, $L=-2$ and $\epsilon =0.4$, does $\delta =0.1$ satisfy the definition?

What is the negation of the claim ${\lim }_{x\to a}f(x)=L$ under the epsilon-delta definition?

For $f(x)=x^2$ at $a=3$ with $L=9$, why use $\delta =\min \left(1,\frac{\epsilon }{7}\right)$ in a proof?

For $f(x)=x^2$ at $a=-1$, using $|x+1|<1$ gives $|x-1|<3$. Which $\delta$ works for every $\epsilon >0$?