Linear functions

Here we want f(x) = x for example. We want to say $\lim_{x\rightarrow \pi} f(x) = \pi$. So for every t > 0 we have to produce m_t > 0 so that if $0 < \vert x-\pi\vert < m_t$ then $\vert f(x) - \pi \vert < t$. This is not hard either:

$$\vert f(x) - \pi\vert = \vert x - \pi\vert,$$

so m_t could be any positive real number less than or equal to t. We can take m_t = t.

Now look at something not quite so obvious: Here we want f(x) = 2x for example. We want to say $\lim_{x\rightarrow \pi} f(x) = 2\pi$. So for every t > 0 we have to produce m_t > 0 so that if $0 < \vert x-\pi\vert < m_t$ then $\vert f(x) - \pi \vert < t$.

$$\vert f(x) -2\pi\vert = \vert 2x - 2\pi\vert = 2\vert x-\pi\vert.$$

Now m_t = t does not work, since f magnifies everything by a factor of 2. Thus we should compensate by taking m_t any positive number less than or equal to half of t. We can take m_t = t/2.