Why should we learn the limit definition of the derivative when we could just learn shortcut rules for differentiation?

Step-by-step explanation:

I think this is a very essential question to ask. While a lot of people want shortcuts to get things done(i used to too, when I was a student), shortcuts doesn't always cut it short. The reason is just as simple as that of any other subject that deals with theory.

Learning the limit definition gives ample knowledge about how the whole derivation works in general. Yes, quite alright, you can get the basic differentiation done when you don't learn that, but as someone who has gone far in calculus, I'd tell you for free, a time will come during computation where you'd get stuck because you never had that prior knowledge of definition.

Further along the line in calculus, you would be doing derivatives where you'd have to apply your knowledge of the definition of limit, and how would you be able get that done if you hadn't already learnt it?

Exactly!

b: |12 – x| ≤ 0.25

step-by-step explanation:      

let x be the possible lengths in inches, of the ribbons jaime cut.

we have been given that jaime is cutting strips of ribbon for a project. he tries to cut each piece to be 12 inches long, but sometimes he cuts it too long or too short by as much as 0.25 inch.  

we can represent this information in an inequality as:

by the definition of absolute value function we will get,

upon combining our inequalities we will get,

therefore, the inequality   describes all the possible lengths (x), in inches, of the ribbons jaime cut.

step-by-step explanation: