Question

Given that lim x→1 (5x − 3) = 2, illustrate definition 2 by finding values of δ that correspond to ε = 0.1, ε = 0.05, and ε = 0.01.

Answer 1

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

is to say that for any $\varepsilon>0$, we can find $\delta>0$ such that $0$ implies $|(5x-3)-2|$.

To that end, we have

$|(5x-3)-2|=|5x-5|=5|x-1|$

which means that if $\varepsilon=0.1$ is given, then we can choose $\delta=\dfrac{0.1}5=0.02$; if $\varepsilon=0.05$, then $\delta=\dfrac{0.05}5=0.01$; if $\varepsilon=0.01$, then $\delta=\dfrac{0.01}5=0.002$.