Epsilon-Delta Definition of a Limit Worked Examples Cheat Sheet

Key Facts

• The formal definition is $\lim_{x\to a} f(x)=L$ if for every $\varepsilon>0$ there exists $\delta>0$ such that $0<|x-a|<\delta$ implies $|f(x)-L|<\varepsilon$.
• For a linear function $f(x)=mx+b$, the limit proof often uses $|f(x)-L|=|m||x-a|$, so choosing $\delta=\frac{\varepsilon}{|m|}$ works when $m\ne0$.
• For $\lim_{x\to a} x^2=a^2$, use $|x^2-a^2|=|x-a||x+a|$ and bound $|x+a|$ by first forcing $|x-a|<1$.
• A common quadratic choice is $\delta=\min\{1,\frac{\varepsilon}{2|a|+1}\}$ because $|x+a|<2|a|+1$ when $|x-a|<1$.
• For a rational function, factor or combine fractions first, then use a restriction such as $|x-a|<1$ to keep denominators away from $0$.
• The condition $0<|x-a|$ means $x$ may approach $a$ but does not have to equal $a$, so the function value $f(a)$ may be undefined.
• The number $\delta$ is allowed to depend on $\varepsilon$, but it must be positive and chosen before assuming $0<|x-a|<\delta$.
• Using $\delta=\min\{c,g(\varepsilon)\}$ lets one condition control nearby behavior while another condition guarantees $|f(x)-L|<\varepsilon$.

Vocabulary

Epsilon

Epsilon, written $\varepsilon$, is a positive tolerance for how close $f(x)$ must be to the limit value $L$.

Delta

Delta, written $\delta$, is a positive distance from $a$ that controls how close $x$ must be to $a$.

Limit

The limit $\lim_{x\to a} f(x)=L$ means $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$.

Punctured neighborhood

A punctured neighborhood of $a$ is the set of points satisfying $0<|x-a|<\delta$.

Bounding

Bounding is the process of replacing a difficult factor with a simpler upper estimate, such as using $|x+a|<2|a|+1$.

Minimum choice

A minimum choice such as $\delta=\min\{1,\frac{\varepsilon}{5}\}$ enforces multiple inequalities at the same time.