Limit vs Derivative
I am learning Calculus and I am confused over the difference between Limit and Derivative.
In the screenshot below, both the blue and red boxes come to the same conclusion of 3x^2.
So, why do we have 2 representations of the same outcome? As far as I understand, the derivative allows us to find the slope of the tangent line, and it seems that the limit equation does the same thing as well.
$\endgroup$ 13 Answers
$\begingroup$Note that
- the derivative at a point $x_0$ is defined by the limit (whether it exists) $$f'(x_0)=\lim_{\Delta x\to 0} \frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}=3x_0^2$$
once we have defined the derivative at each point we can consider its value for all points of the domain such that the given limit exists and thus define the derivative function $f'(x)=3x^2$.
$\endgroup$ 8 $\begingroup$The concepts of a limit and derivative are totally different. One represents the idea of moving infinitely close to some input of a function and observing its behavior. Informally, a derivative is what happens when you measure the rate of change between two infinitely close points.
In just so happens that someone is using the definition of a derivative to show that a certain limit is equal to the derivative of some function. The derivative is built upon the limit and is itself a limit:
$$\lim_{h\to 0} {f(x+h)-f(x)\over h} = f'(x)$$ $$\lim_{x\to a} {f(x)-f(a)\over x-a} = f'(x)$$
The two are inherently related because limits are a mathematical tool to encapsulate what we are actually doing with a derivative, but they are not the same conceptually.
$\endgroup$ $\begingroup$The derivative is the limit formula. More so, the derivative is defined as$$\frac {df}{dx}=\lim\limits_{\Delta x\to0}\frac {f(x+\Delta x)-f(x)}{\Delta x}$$The reason why they produce the same answer is because they are the exact same formula. The derivative is calculated using the limit.
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