[Math] Limit of a function when x approaches infinity

calculusfunctionslimits

I need to prove the following limit using definition only:
$$\lim_{x\to -\infty} \frac{(7x+3)}{(x-1)} =7.$$

The definition is: for any $\epsilon >0$. there is a $\delta$ so $x<\delta \implies |f(x)-L|<\epsilon$

In order to show that $|f(x) – L| < \epsilon$ I assumed $\delta =1, \delta=0$ and in the end i showed $\delta= 10/\epsilon$. The problem is I don't know which one to choose: $\min\{\delta_1, \delta_2, \cdots\}$ or $\max\{\delta_1, \delta_2, \cdots\}$. In the normal definition of a limit i know that we always choose the minimum delta but i am not sure what do here.

My second question is: can I assume many things about $\delta$ then just choose minimum/maximum, is it ok?

Thanks for help.

Best Answer

What we want in the definition is

For any $\epsilon > 0$, there is a $\delta$ such that $x \lt \delta \implies |f(x) - L| \lt \epsilon.$

(Definition since corrected in the OP).

For any $\epsilon > 0$, we can make $\delta$ as small as we'd like to ensure the implication holds, and in the end, we select the minimum $\delta$.

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