Vertical Tangent Lines (2)
 
After learning how to handle cases where vertical tangent lines accompany the graph of a function, we also run into cases where the analysis of the slope of a tangent line gives yet another clue about the shape of a graph as the graph approaches a point where the function is NOT even defined.
 
The example below presents such a case. Notice that the function f (x) = x·ln(x) is NOT even defined at x = 0. Yet, the analysis of the slope of a tangent line gives critical information on how the graph should behave near x = 0.
  

 
The derivative in this problem is y' = ln(x) + 1. Relying on what we know about the graph of ln(x), we see that the values of y' approach –∞ as x approaches 0 while x > 0. Geometrically, this means that a tangent line becomes vertical. Equivalently, the graph should become vertical as it approaches the origin (0, 0) from the right.
 
Do NOT conclude that the graph has a vertical tangent line x = 0. Why? Because the function is undefined at x = 0, hence the point (0, 0) is NOT on the graph. We should draw an open circle at the origin.