Write an epsilon delta proof for the expression $\lim_{x \to - \infty} (3x - 5) = - \infty$

Question

I believe I have the right answer for this question however after looking at the answer I am a little confused.

heres my work:

$3x - 5 < -M$ at the same time x < -N

x < $\frac{5 - M}{3}$

$\frac{5 - M}{3}$ < -N

N < $\frac{M - 5}{3}$

So i feel I've clearly shown how $N = \frac{M - 5}{3}$ . What I'm not understanding is why the actual answer takes N to be the max of $(0, \frac{M - 5}{3})$ why do we take $0$ if $M < 5$.

Answer 1

I suspect that the definition of $\displaystyle\lim_{x\to-\infty}f(x)=-\infty$ that you are working with is$$(\forall M\color{red}{\geqslant0})(\exists N\color{red}{\geqslant0}):x<-N\implies f(x)<-M.$$So, as you see, it's part of the definition that $N>0$. That's why, in your example, you take $N=\max\left\{0,\frac{M-5}3\right\}$: to make sure that $N\geqslant0$.