Weak tangent but not a strong tangent

Weak tangent

Notice that the slope of $(\lambda h^3, \lambda h^2)$ tends to $\infty$ as $h\to 0$, meaning that the line direction approaches vertical. Since the line always passes through $(0,0)$, this means it has a limiting position (the $y$ axis).

Strong tangent

If it exists, it has to be the same as the weak tangent, because if the double limit exists, iterated limit "$k\to 0$ then $h\to 0$" exists and is equal to it. However, approaching via $h=-k$ you will find that the lines stay horizontal.

Intuitive meaning

• Strong tangent: if you walk along the curve and someone is walking along the tangent line with the same speed, you can spend some time walking together and holding hands.

• Weak tangent: looks like strong tangent at first, but at the point of tangency there is a break-up and someone goes away in the opposite direction.