If a function f is continuous at a and \(\lim _{x \rightarrow a}\ |f'(x)|=\infty\), then 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.

Graph the following curves and determine the location of any vertical tangent lines.

a. \(x^2+y^2=9\) b. \(x^2+y^2+2x=0\)

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 2 2 (a). Given the function x +y = 9 2 We can write this function as y =± 9x Now,let’s differentiate this function y, 2x x Therefore y = 2 9x = 9x2 Now let us take the left hand and the right hand limits. lim|f(x)| = lim| x |= x3 x3 9x2 lim |f(x)| = lim | x 2|= x3 x3 9x Therefore,we have lim|f (x)| = lim |f(x)| = x3 x3 Thus the given function has vertical tangent lines at x=3 and x=-3 STEP 2 (b). Given the function x +y +2x = 0 We can write this function as y =± x 2x 2 Now,let’s differentiate this function y, Therefore y = 2x2 = x1 = (x+1) 2x 2x x 2x x 2x Now let us take the left hand and the right hand limits. lim|f(x)| = lim| x+1 |= x0 x0 x 2x lim |f(x)| = lim | x+1 |= x2 x2 x 2x Therefore,we have lim|x0 (x)| =x2 |f(x)| = Thus the given function has vertical tangent lines at x=0 and x=-2