No Tangent Line: Graph of y = |x| Explained

[Moneer81]

Am I thinking right about this following question:

Give a geometric argument to show that the graph of y = lxl has no tangent line at the point (0,0).

Since that graph would be basically the bisectors of the 1st and second quadrants, why can't the x-axis be that tangent line?
(def. of a tangent line is a line that touches the graph at one point, which is the origin in this case)

thanks

 

[d_leet]

The slightly more correct definition of a tangent line at a point is the limit of secant lines through that point and one near it, as the nearer point approaches infinitely close to the point you want to find the tangent line to. If you do this for the function f(x) = |x| you'll notice what happens when you take points on both sides of the origin to draw secant lines through.

 

[Moneer81]

so the x-axis doesn't qualify as a tangent line?

 

[d_leet]

so the x-axis doesn't qualify as a tangent line?

In a sense it is, but it doesn't fit the definition of a tangent line that they want you to use. I explained what they really want you to do in my previous post, and if you think about that you will see why there is no tangent line at x = 0.

 

[HallsofIvy]

(def. of a tangent line is a line that touches the graph at one point, which is the origin in this case)

If that were the definition of tangent line, then every straight line through the origin would be a tangent line there. Check your textbook for the specific definition of "the tangent line to a graph".