Problem

Can you find a tangent line to f(x) = |x| at x = 0?

Solution

To see whether this is possible, we will try to determine what the slope of the tangent line would be. From the book, we know that the slope of the tangent at x = 0 is defined to be the limit

if this limit exists.

If we aproach the limit from the right,

If we approach the limit from the left,

The two-sided limit does not exist, since the one-sided limits have different values. Therefore, there is no definition for the slope of a tangent line at zero, so there is no tangent line to |x| at zero.