Vertical tangent lines? If a function f is continuous at a and lim|f (x)| = ? , then x?a the curve y= f(x) has a vertical tangent line at a and the equation of the tangent line is x = a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 59-60) is used. Use this definition to answer the following questions. The preceding definition of vertical tangent line includes four cases: x?am+f ?x) =± ? combined with lim f (xx?a±???(for example, one case is lim f ?x) =+ ?and lim f (x) =? ?). Make a rough sketch of a (continuous) x?a + x?a ? function that has a vertical tangent line at x = a in each of the four cases.

SOLUTION If a function f is continuous at a and lim|fxa (x)| = , then the curve y= f(x) has a vertical tangent line at a and the equation of the tangent line is x = a STEP 1 We need to sketch a graph of a continuous function that has a vertical tangent line at x = a in each of the four cases. Since a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency....