Processing math: 18%

EXAMPLE 5Using the ϵ-δ Definition of a Limit

Use the ϵ-δ definition of a limit to prove the statement lim.

Solution We use a proof by contradiction.* Assume \lim\limits_{x\rightarrow 3}(4x-5)=10 and choose \epsilon =1. (Any smaller positive number \epsilon will also work.) Then there is a number \delta >0, so that \begin{equation*} \hbox{whenever }\quad 0\lt\vert x-3\vert \lt\delta\qquad \hbox{then}\quad \vert ( 4x-5) -10\vert \lt1\ \end{equation*}

We simplify the right inequality. \begin{equation*} \begin{array}{rcl} \vert (4x-5)-10 \vert &=& \vert 4x-15 \vert \lt1 \\[3pt] -1&\lt&4x-15\lt 1 \\[3pt] 14&\lt&4x\lt 16 \\[3pt] 3.5&\lt&x\lt 4 \end{array} \end{equation*}

*In a proof by contradiction, we assume that the conclusion is not true and then show this leads to a contradiction.

For example, if \delta =\dfrac{1}{4}, then 3-\dfrac{1}{4}\lt x\lt 3+\dfrac{1}{4}\\ 2.75\lt x\lt 3.25 contradicting {3.5\lt x \lt4}.

According to our assumption, whenever 0\lt\left\vert x-3\right\vert \lt\delta , then 3.5\lt x\lt 4. Regardless of the value of \delta, the inequality 0\lt \left\vert x-3\right\vert \lt \delta is satisfied by a number x that is less than 3. This contradicts the fact that 3.5\lt x\lt 4. The contradiction means that \lim\limits_{x\rightarrow 3}(4x-5)\neq 10.