Why does l'hopital's rule work?

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What is the intuition behind it? when i watch videos of people using l'hopital's rule. i can only deduce that they're only taking derivatives over and over again until a number comes out and that becomes the limit. how can a tangent slope be a value for a limit? Please give me an intuitive explanation. I'm a novice with abstract explanations. thank you!
 

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Ssnow
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PeroK
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If ##f(a) = g(a) = 0## then it's as simple as:

##lim_{x \rightarrow a} \frac{f(x)}{g(x)} = lim_{x \rightarrow a} \frac{(f(x) - f(a)/(x-a)}{(g(x) - g(a))/(x-a)} = \frac{f'(a)}{g'(a)}##

That's the essence of it, anyway.
 
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If ##f(a) = g(a) = 0## then it's as simple as:

##lim_{x \rightarrow a} \frac{f(x)}{g(x)} = lim_{x \rightarrow a} \frac{(f(x) - f(a)/(x-a)}{(g(x) - g(a))/(x-a)} = \frac{f'(a)}{g'(a)}##

That's the essence of it, anyway.
wow! fascinating! lol...
 
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If ##f(a) = g(a) = 0## then it's as simple as:

##lim_{x \rightarrow a} \frac{f(x)}{g(x)} = lim_{x \rightarrow a} \frac{(f(x) - f(a)/(x-a)}{(g(x) - g(a))/(x-a)} = \frac{f'(a)}{g'(a)}##

That's the essence of it, anyway.
This formula is intuitive and really drives home what a derivative actually is. Thanks for sharing it.
f(x) - f(a) is really a fancy way of saying change of y, denoted as Δy, or in other words, y2 - y1. Similarly, x - a is a fancy way of writing change in x, denoted Δx, or in other words x2 - x1.
As "x" approaches "a" the change in x gets smaller, and so does the change in y. When we had to find slopes in algebra we had to use a point-slope form, which was (y2-y1) / (x2-x1). We can apply this same concept to non-linear functions, and when we make "x" really close to "a" then we get a more accurate approximation of the slope of the curve. Furthermore, a first derivative is the SLOPE of the line tangent to a function, and where this tangent line touches the function the two functions are said to have the same slope. In other words, when a=x we then have an instantaneous rate of change; a derivative. In short, I love all the implications from this formula.

I am not sure why L'Hopital's rule works, but I do know that you must first observe a limit to yield an indeterminate form (0/0, ∞/∞, etc) before you can apply L'Hopital's rule. The formula PeroK provided is interesting because it makes a ratio of the instantaneous rate of change of function f to the instantaneous rate of change of function g at the same point "a". Maybe because f(x) divided by g(x) is a ratio, looking at the slopes of each function (and the ratio of the slopes at a particular "a" value) can provide logical insights. When I think of L'Hopital's Rule, I can't help but think of Taylor series (which is a summation of derivatives (of derivatives) and corresponding polynomials, which approximate a given function at a value x near the center of the function). Although a strange thought, can anyone relate Taylor series to L'Hopital's Rule?
 
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