From: 3blue1brown
The concept of a limit is fundamental in calculus, helping to define what it means for one value to “approach” another [00:00:25]. It’s essentially a matter of assigning fancy notation to the intuitive idea of one value getting closer to another [00:00:29]. Limits allow mathematicians to make intuitive ideas more airtight and rigorous [00:08:16].
Limits and the Definition of a Derivative
The idea of a derivative heavily relies on limits [00:00:14]. When considering a function f(x), its derivative at a point (e.g., x = 2) involves imagining a small nudge dx to the input and observing the resulting change df in the output [00:01:29]. The ratio df / dx represents the rise over run slope [00:01:37]. The actual derivative is whatever this ratio approaches as dx approaches 0 [00:01:49].
The formal definition of a derivative is expressed using limits:
lim (h → 0) [f(x + h) - f(x)] / h [00:03:01].
Here, h (or delta x) represents the finite, non-zero nudge to the input, avoiding the paradoxical idea of an infinitely small change [00:03:11]. The use of lowercase d terms like dx in typical derivative expressions implicitly contains this idea of a limit, where dx is supposed to eventually go to 0 [00:03:35].
Understanding “Approach”
The epsilon-delta definition of limits provides a rigorous way to understand what “approach” means [00:01:08]. For a limit to exist, for any desired small distance epsilon around the limiting output value, there must be a corresponding input range delta around the limiting input point such that all outputs within that delta range are within epsilon of the limiting value [00:08:46]. If no such delta can be found for a given epsilon (meaning the output range can’t be made arbitrarily small), then the limit does not exist [00:07:15].
The Problem: Indeterminate Forms
Sometimes, when evaluating a function at a specific input, direct substitution leads to an indeterminate form like 0/0 [00:05:32]. For example, consider the function (sin(pi * x)) / (x^2 - 1) [00:10:12]. If you try to plug in x = 1, both the numerator sin(pi) and the denominator (1^2 - 1) become 0, making the function undefined at that point [00:10:30]. However, the graph might appear to approach a distinct value as x approaches 1 [00:10:50].
One way to approximate this value is to plug in a number very close to 1, like 1.00001 [00:11:07]. This might suggest a value around -1.57 [00:11:16]. But to find the precise limit, a systematic process is needed [00:11:21].
L’Hopital’s Rule: A Clever Trick
After limits are used to define derivatives, derivatives can, in turn, help evaluate limits [00:11:36]. This is where L’Hopital’s Rule comes in [00:16:13].
Derivation/Intuition
Consider two functions, f(x) and g(x), both equal to 0 at some common value x = a [00:14:24]. When examining the ratio f(x) / g(x) as x approaches a, a tiny nudge dx away from a can be considered [00:15:01].
- The value of
fata + dxis approximately its derivativef'(a)multiplied bydx[00:15:06]. - Similarly, the value of
gata + dxis approximately its derivativeg'(a)multiplied bydx[00:15:15].
Therefore, near the “trouble point” a, the ratio f(x) / g(x) is approximately (f'(a) * dx) / (g'(a) * dx) [00:15:25]. The dx terms cancel out, meaning the ratio of f and g near a is approximately the same as the ratio between their derivatives evaluated at a [00:15:37]. Since these approximations become more accurate for smaller dx, this ratio of derivatives gives the precise value for the limit [00:15:45].
Applying the Rule
L’Hopital’s Rule states that if lim (x → a) f(x) = 0 and lim (x → a) g(x) = 0, then:
lim (x → a) [f(x) / g(x)] = lim (x → a) [f'(x) / g'(x)] [00:15:58].
For the example (sin(pi * x)) / (x^2 - 1) as x approaches 1:
- Numerator derivative: The derivative of
sin(pi * x)iscos(pi * x) * pi(by the chain rule) [00:12:33]. Atx = 1, this iscos(pi) * pi = -1 * pi = -pi[00:13:03]. - Denominator derivative: The derivative of
x^2 - 1is2x[00:13:12]. Atx = 1, this is2 * 1 = 2[00:13:29].
Applying L’Hopital’s Rule, the limit is (-pi) / 2 [00:13:50]. This tells us the precise limiting value for the original function [00:14:02].
Important Considerations
- When to use: L’Hopital’s Rule is useful whenever an expression evaluates to an indeterminate form (like 0/0) when you plug in a particular input [00:15:58].
- Limitations: It cannot be used to discover new derivative formulas because applying it would require already knowing the derivative of the numerator [00:16:45].
- History: Although known as L’Hopital’s Rule, it was actually discovered by Johann Bernoulli, who was paid by L’Hopital for the rights to some of his mathematical discoveries [00:16:17].