Existence of Tangent

 


Let   y=f(x)    be any function and

dy/dx   exists at     x=a    

Then at the corresponding point    [a , f(a)]    of the function tangent to the curve exists and if the tangent

makes an angle    theta     with the positive direction of    x-axis    we have

tan(theta)=dy/dx    at     x=a .

tan(theta) = gradient of the tangent to the curve

      y=f(x)     at    x=a 

therefore ; if    f(x)   is differentiable at    x=a    then tangent to the curve at

x=a    must exist and it must be unique .

In any graph of    /x/   tangent is not unique at    x=0   .

i.e. ; tangent at point    x=0   when     x-->0    from left is not same as the tangent

x=0    when   x-->0   from right .

when   x-->0   from left , gradient of the tangent is

tan135degree=-1    and when   x-->0  from right , gradient of the tangent is

tan45degree=1.

Hence tangent is not unique at    x=0    and consequently    /x/  is not differentiable at     x=0    .