Epsilon-Delta Practice Problems

In your own words, explain the roles of $\varepsilon$ and $\delta$.

Rewrite $|x-3|<0.2$ as an ordinary interval.

Rewrite $|f(x)-5|<0.1$ as a vertical output interval.

For $f(x)=4x$, find a $\delta$ in terms of $\varepsilon$ that proves $\lim_{x\to2}f(x)=8$.

Prove $\lim_{x\to1}(5x-2)=3$.

Prove $\lim_{x\to-2}(3x+4)=-2$.

Prove $\lim_{x\to4}(x/2)=2$.

Prove $\lim_{x\to1}x^2=1$ using $\delta=\min\{1,\varepsilon/3\}$.

Prove $\lim_{x\to3}x^2=9$ by first bounding $|x+3|$.

Find a suitable $\delta$ to prove $\lim_{x\to2}(x^2+x)=6$.

Explain why a proof may choose a smaller $\delta$ than necessary.

Explain why $\delta$ may depend on $\varepsilon$, but not on the particular $x$ chosen later.

Use $\eps_0=1/4$ to show the step function in the chapter does not approach $1/2$ at zero.

Give a formal $M$-$\delta$ proof that $1/x^2\to+\infty$ as $x\to0$.

Give a formal $N$-style proof that $1/x\to0$ as $x\to\infty$.

Identify the first incorrect step in the claim: "Choose $\delta=\varepsilon$. Then $|x^2-4|<\varepsilon$ whenever $|x-2|<\delta$."

Why is $0<|x-a|$ included in the formal finite-limit definition?

Explain how the graphical $\varepsilon$-band and $\delta$-window represent the quantified definition.