In words, what does \(\lim_{x\to a}f(x)=L\) mean?

In the epsilon-delta definition, which quantity is chosen first: \(\epsilon\) or \(\delta\)?

Translate \(|f(x)-L|<\epsilon\) into interval notation for \(f(x)\).

Translate \(0<|x-a|<\delta\) into a statement about \(x\).

Use the epsilon-delta definition to prove \(\lim_{x\to2}(x+3)=5\).

Use the epsilon-delta definition to prove \(\lim_{x\to1}2x=2\).

Find a suitable \(\delta\) in terms of \(\epsilon\) to prove \(\lim_{x\to4}(3x-1)=11\).

For \(f(x)=5-2x\), find \(\delta\) in terms of \(\epsilon\) to prove \(\lim_{x\to1}f(x)=3\).

Prove \(\lim_{x\to3}(4x+1)=13\) using epsilon-delta language.

Prove \(\lim_{x\to-2}(7-x)=9\) using epsilon-delta language.

Explain why the condition \(0<|x-a|\) appears in the formal limit definition.

A function satisfies \(f(x)=2x+1\) for \(x\ne1\), but \(f(1)=100\). What is \(\lim_{x\to1}f(x)\), and why?

Prove \(\lim_{x\to2}x^2=4\) by first restricting \(|x-2|<1\).

Prove \(\lim_{x\to1}x^2=1\) using an epsilon-delta argument.

Find a \(\delta\) that works for \(\lim_{x\to0}x^2=0\).

For \(\lim_{x\to a}cx=ca\), where \(c\ne0\), find a suitable \(\delta\) in terms of \(\epsilon\).

A proof chooses \(\delta=\epsilon^2\) to prove \(\lim_{x\to0}x=0\). Does it work for every \(\epsilon>0\)? Explain.

Disprove the claim \(\lim_{x\to0}1=2\) using the epsilon-delta definition.

Explain why the order \(\forall\epsilon>0\,\exists\delta>0\) cannot be swapped in the formal definition of a limit.

Prove \(\lim_{x\to a}(mx+b)=ma+b\) for constant \(m\ne0\).