Quiz:Limit and continuity

Formal definition of limit and continuity

1 Which of these is the correct interpretation of $\underset{x\to c}{lim}f(x)=L$ in terms of the definition of limit?

	For every $\alpha >0$, there exists $\beta >0$ such that if $0<|x-c|<\alpha$, then $|f(x)-L|<\beta$.
	There exists $\alpha >0$ such that for every $\beta >0$, and $0<|x-c|<\alpha$, we have $|f(x)-L|<\beta$.
	For every $\alpha >0$, there exists $\beta >0$ such that if $0<|x-c|<\beta$, then $|f(x)-L|<\alpha$.
	There exists $\alpha >0$ such that for every $\beta >0$ and $0<|x-c|<\beta$, we have $|f(x)-L|<\alpha$.
	None of the above

2 Suppose $f:\mathbb{R}\to \mathbb{R}$ is a function. Which of the following says that $f$ does not have a limit at any point in $\mathbb{R}$ (i.e., there is no point $c\in \mathbb{R}$ for which $\underset{x\to c}{lim}f(x)$ exists)?

	For every $c\in \mathbb{R}$, there exists $L\in \mathbb{R}$ such that for every $ϵ>0$, there exists $\delta >0$ such that for all $x$ satisfying $0<|x-c|<\delta$, we have <math>|f(x) - L| \ge
	There exists $c\in \mathbb{R}$ such that for every $L\in \mathbb{R}$, there exists $ϵ>0$ such that for every $\delta >0$, there exists $x$ satisfying $0<|x-c|<\delta$ and $|f(x)-L|\ge ϵ$.
	For every $c\in \mathbb{R}$ and every $L\in \mathbb{R}$, there exists $ϵ>0$ such that for every $\delta >0$, there exists $x$ satisfying $0<|x-c|<\delta$ and $|f(x)-L|\ge ϵ$.
	There exists $c\in \mathbb{R}$ and $L\in \mathbb{R}$ such that for every $ϵ>0$, there exists $\delta >0$ such that for all $x$ satisfying $0<|x-c|<\delta$, we have $|f(x)-L|\ge ϵ$.
	All of the above.

3 In the usual $ϵ-\delta$ definition of limit for a given limit $\underset{x\to c}{lim}f(x)=L$, if a given value $\delta >0$ works for a given value $ϵ>0$, then which of the following is true?

	Every smaller positive value of $\delta$ works for the same $ϵ$. Also, the given value of $\delta$ works for every smaller positive value of $ϵ$.
	Every smaller positive value of $\delta$ works for the same $ϵ$. Also, the given value of $\delta$ works for every larger value of $ϵ$.
	Every larger value of $\delta$ works for the same $ϵ$. Also, the given value of $\delta$ works for every smaller positive value of $ϵ$.
	Every larger value of $\delta$ works for the same $ϵ$. Also, the given value of $\delta$ works for every larger value of $ϵ$.
	None of the above statements need always be true.

4 Which of the following is a correct formulation of the statement $\underset{x\to c}{lim}f(x)=L$, in a manner that avoids the use of $ϵ$s and $\delta$s?

	For every open interval centered at $c$, there is an open interval centered at $L$ such that the image under $f$ of the open interval centered at $c$ (excluding the point $c$ itself) is contained in the open interval centered at $L$.
	For every open interval centered at $c$, there is an open interval centered at $L$ such that the image under $f$ of the open interval centered at $c$ (excluding the point $c$ itself) contains the open interval centered at $L$.
	For every open interval centered at $L$, there is an open interval centered at $c$ such that the image under $f$ of the open interval centered at $c$ (excluding the point $c$ itself) is contained in the open interval centered at $L$.
	For every open interval centered at $L$, there is an open interval centered at $c$ such that the image under $f$ of the open interval centered at $c$ (excluding the point $c$ itself) contains the open interval centered at $L$.
	None of the above.