Vertical tangent lines? ?If a function f is continuous at?a and , ?then th?e ?cur? ?y=? (?x) ?has a vertical tangent line at a and the equation of the tangent? line? is x =? . ?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. ? ? eri? that ? ? ) = ?x1/3 has a vertical tan?gent line at ? = 0.

SOLUTION If a function f is continuous at a and lim|f (x)| = , then the curve y= f(x) has a xa vertical tangent line at a and the equation of the tangent line is x = a STEP 1 Given f(x) = x 3 1 11 1 2 Then f (x) = 3 3 = 3 3 STEP 2 Now let us take the left hand and the right hand limits. 1 2 lim+|f(x)| = lim + 3 |= x0 x0 1 32 lim|f(x)| = lim 3 |= x0 x0 Therefore,we have lim +f(x)| = lim f(x)| = x0 x0 Thus the given function has a vertical tangent line at x=0 STEP 3 The graph of the given function is