Question

How do derivatives relate to limits?

Answer 1

Here in this question, we have to explain what is the relation between the limits and derivatives. For this we have to know the definitions of both limit and derivatives and next we explain how limits influence or on the derivatives or vice versa. Let us discuss one by one in the below section.

Complete step by step solution:
Historically, Sir Issac Newton was the "inventor" of derivatives and Leibnitz introduced the concept of Limits. Limits and derivatives are extremely crucial concepts in Maths whose application is not only limited to Maths but are also present in other subjects like physics.
A limit is defined as a value that a function approaches as the input, and it produces some value. Limits are important in calculus and mathematical analysis and used to define integrals, derivatives, and continuity.
To express the limit of a function, we represent it as: $\mathop {\lim }\limits_{n \to c} f\left( n \right) = L$
A derivative refers to the instantaneous rate of change of a quantity with respect to the other. It helps to investigate the moment by moment nature of an amount. The derivative of a function is represented in the below-given formula.
$\mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}$
The relation between a limit and a derivative if you look at their definition. First, the concept of limit is defined:
$\mathop {\lim }\limits_{x \to \alpha } f\left( x \right) = l\,\,\,\, \Leftrightarrow \,\,\,\,\forall U\left( l \right),\,\,\exists \,\,V\left( \alpha \right)$ such that $x \ne \alpha \in V\left( \alpha \right)\,\, \Rightarrow \,\,f\left( x \right) \in U\left( l \right)$
where the notation $U\left( {{x_0}} \right)$ denotes the neighbourhood of ${x_0}$. Also note that in your example the limit to ${x_0}$ is just $f\left( {{x_0}} \right)$, but this only holds for continuous functions. In general, we are not interested in the function's behaviour at the point we are evaluating the limit. This is reflected in the definition, as the $x \ne \alpha$ shows.
After we define what a limit represents you can define derivatives:
$f'({x_0}) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {{x_0} + h} \right) - f\left( {{x_0}} \right)}}{h} \in \mathbb{R}$
As we can see a derivative is defined on top of the limit concept, so this is the relation between the two.

Note: The question is a descriptive type of the question. The limit concept first came into existence and then after the limit the derivative came into existence. So the derivative can be written in the form of the limit is given as $f'({x_0}) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {{x_0} + h} \right) - f\left( {{x_0}} \right)}}{h} \in \mathbb{R}$