Two young mathematicians discuss linear approximation.

Check out this dialogue between two calculus students (based on a true story):
Devyn
Hmmmm. Riley, I just thought of something…
Riley
What is it?
Devyn
When we compute derivatives, we are looking at the slope of tangent lines right?
Riley
You know it.
Devyn
Well, I wonder: Instead of studying curves, could we just study “zoomed-in” lines?
Riley
I’m not sure…

You read someplace that is a good approximation for when is close to .

Plot and . Explain how this shows that is a good approximation when is close to .
Explain (if you can) using concepts of calculus to explain why is a good approximation for when is close to .