Infinite non deviating slope

Question

Okay, I had a question that my math teacher didn't know the answer to, and that I haven't found an answer for on the web. Say you are graphing a system of equations, right, and you have (hypothetically) two slopes of 2, and 2 and one infinitieth, both starting at the origin. Since it would take an infinitely long time for the two lines to deviate, could they be said to coincide? I understand they're different numbers, but if you can never reach the end of infinity, therefore never deviating from the other line, wouldn't they be the same? Any help would be appreciated. Thanks.

Answer 1

In the standard reals, infinitesimals do not exist. What corresponds to this idea is $\lim_{c \to 0} f(c)$ for some function $f$.

In your case, consider the lines $y=2x$ and $y=(2+c)x$ and let $c \to 0$.

For any particular $c$, the two lines differ at $x$ by $cx$. You can therefore say two things:

1. For any fixed value of $c$, the two line get arbitrarily far apart for large $x$.

2. For any fixed value of $x$, the two lines can be made arbitrarily close at $x$ by choosing $c$ small enough.

As to what happens when both $x$ gets large and $c$ gets small, it depends on how that happens.