We need to remove fear. But mothers fear for their kids? If I have no expectation, I can't feel fear. I can't be afraid of dying. Yet, when I fall off a bike, isn't it fear that protects me from dying? Mothers also often worry about their kids. Are they weak for doing so?
In Buddhism, there are different kinds of fear or 'good fear' &amp; 'bad fear'. 

Bad fear is fearing what should not be feared (e.g. afraid of dying when you are old or terminally ill or fearing what other people think about you).

Good fear called 'ottappa' is fearing what should be feared; such as fearing doing bad actions that will bring a bad results (for example, fearing addiction to drugs). 

The Dhammapada states:


  317. Those who see something to fear where there is nothing to fear and see nothing to fear where there is something to fear — upholding
  false views, they go to states of woe.