What is L’Hospital’s Rule?
L'Hôpital's Rule, named after the French mathematician Guillaume de l'Hôpital, is a mathematical theorem that helps to evaluate certain indeterminate forms when taking the limit of a quotient of two functions. These indeterminate forms often arise in calculus when you encounter limits of the form 0/0 or ∞/∞. The rule states that if the limit of the ratio of two functions is in an indeterminate form, and if both the numerator and denominator are differentiable at a particular point, then you can evaluate the limit by taking the derivative of the numerator and the derivative of the denominator and then re-evaluating the limit.
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L'Hôpital's Rule is typically expressed as follows:
If lim(x → a) [f(x) / g(x)] = 0/0 or ∞/∞, and if both f(x) and g(x) are differentiable in an open interval containing x = a (except possibly at x = a), then:
- lim(x → a) [f(x) / g(x)] = lim(x → a) [f'(x) / g'(x)]
In simpler terms, you can find the limit of the ratio of the derivatives of the functions in the numerator and denominator when you encounter indeterminate forms, 0/0 or ∞/∞, in a limit calculation.
L'Hôpital's Rule is a useful tool for solving certain limits and can be applied repeatedly if the indeterminate form persists after the initial application. It is often used in calculus when working with functions that have singularities, asymptotes, or other challenging behavior near a particular point.
L’Hospital’s Rule Formula
L'Hôpital's Rule, named after the French mathematician Guillaume de l'Hôpital, is a mathematical technique for evaluating the limit of an indeterminate form (0/0 or ∞/∞) when you have a quotient of two functions. The rule can be stated as follows:
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"If you have a limit of the form:
lim (x → c) [f(x) / g(x)]
and both f(x) and g(x) approach 0 (or ∞) as x approaches c, then you can apply L'Hôpital's Rule. The rule states that if the limit of the quotient of the derivatives of f(x) and g(x) exists as x approaches c, then this limit is equal to the limit of the original function. In other words:
lim (x → c) [f(x) / g(x)] = lim (x → c) [f'(x) / g'(x)]
Here, f'(x) and g'(x) represent the derivatives of f(x) and g(x) with respect to x, respectively.
You can apply L'Hôpital's Rule repeatedly if necessary, until you reach a limit that can be easily evaluated. It's important to note that L'Hôpital's Rule is only applicable in cases where you have an indeterminate form (0/0 or ∞/∞) and both the numerator and denominator approach the same limit as x approaches c."
L’Hospital’s Rule Proof
L'Hôpital's Rule, named after the French mathematician Guillaume de l'Hôpital, is a powerful tool for finding the limit of an indeterminate form, particularly when dealing with limits involving fractions. The indeterminate forms it addresses include 0/0 and ∞/∞.
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Here's a proof of L'Hôpital's Rule for the case when we have a limit of the form 0/0:
Theorem (L'Hôpital's Rule): Let f(x) and g(x) be continuous functions on an open interval containing a, except possibly at a, and suppose that:
Then, lim (x→a) [f(x) / g(x)] = lim (x→a) [f'(x) / g'(x)], provided the right-hand limit exists.
Proof:
Start with the given conditions: lim (x→a) f(x) = 0 and lim (x→a) g(x) = 0. We want to find lim (x→a) [f(x) / g(x)].
Since both f(x) and g(x) are continuous at a, they are differentiable at a (except possibly for g(x) = 0 at a). Therefore, by L'Hôpital's Rule, we can find the derivatives of f(x) and g(x) at a: f'(a) and g'(a).
Now, consider the limit of the quotient f(x) / g(x) as x approaches a:
lim (x→a) [f(x) / g(x)]
Using the definition of the derivative, we can write f(x) and g(x) as:
f(x) = f(a) + f'(a)(x - a) + ε1(x) g(x) = g(a) + g'(a)(x - a) + ε2(x)
Here, ε1(x) and ε2(x) are functions that tend to 0 as x approaches a. These functions represent the difference between f(x) and g(x) and their tangent lines at the point (a, 0).
Now, rewrite the limit as x approaches a using the definitions from step 4:
lim (x→a) [f(x) / g(x)] = lim (x→a) [(f(a) + f'(a)(x - a) + ε1(x)) / (g(a) + g'(a)(x - a) + ε2(x))]
Divide both the numerator and denominator by (x - a):
lim (x→a) [f(x) / g(x)] = lim (x→a) [(f(a) / (x - a) + f'(a) + ε1(x) / (x - a)) / (g(a) / (x - a) + g'(a) + ε2(x) / (x - a))]
As x approaches a, the terms f(a) / (x - a), g(a) / (x - a), and ε1(x) / (x - a) all approach 0 because they are of the form 0/0. Thus, our limit becomes:
lim (x→a) [f(x) / g(x)] = lim (x→a) [f'(a) / g'(a)]
Since the limit on the right is independent of x and equals the ratio of the derivatives, we've shown that:
lim (x→a) [f(x) / g(x)] = f'(a) / g'(a)
This completes the proof of L'Hôpital's Rule for the case when we have a limit of the form 0/0. The proof for the case when we have a limit of the form ∞/∞ is similar, and the theorem covers both situations.
L'Hopital's Rule History
L'Hôpital's Rule, also known as l'Hôpital's rule or Hospital's rule, is a mathematical theorem used to evaluate the limit of an indeterminate form of a ratio of two functions. It is a fundamental tool in calculus and is often employed when trying to determine the limit of a quotient where both the numerator and denominator tend to zero or both tend to infinity.
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The rule is named after the French mathematician Guillaume François Antoine, Marquis de l'Hôpital. L'Hôpital's Rule is associated with his book "Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes" (Analysis of the Infinitely Small for the Understanding of Curved Lines), which was published in 1696. In this work, l'Hôpital credited Johann Bernoulli, another renowned mathematician of the time, for the rule. The book contains a letter from Bernoulli to l'Hôpital in which he discusses the rule for calculating the limit of a fraction by taking the limit of the derivatives of the numerator and denominator.
L'Hôpital's Rule gained prominence due to the efforts of Bernoulli, who actively promoted it and wrote extensively about it. Although l'Hôpital is associated with the rule, he himself did not derive the rule but rather popularized it based on the work of Bernoulli.
The general statement of l'Hôpital's Rule is as follows:
If the limit of the ratio of two functions f(x)/g(x) as x approaches a particular value c is of an indeterminate form 0/0 or ∞/∞, and both f(x) and g(x) are differentiable at c (with g'(c) ≠ 0), then the limit can be evaluated by taking the limit of the derivatives of f(x) and g(x) as x approaches c:
lim (x → c) [f(x)/g(x)] = lim (x → c) [f'(x)/g'(x)]
This rule provides a systematic method for finding limits in cases where direct substitution results in an indeterminate form. L'Hôpital's Rule is an essential tool in calculus and is widely taught and used in mathematics and science.
L’Hospital’s Rule Uses
L'Hôpital's Rule, named after the French mathematician Guillaume de l'Hôpital, is a technique used to evaluate the limit of an indeterminate form of a fraction, specifically when the limit of the numerator and the limit of the denominator both approach zero or both approach infinity. The indeterminate forms it addresses are typically 0/0 or ∞/∞.
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The rule states that if you have a limit of the form:
Lim (x → c) [f(x) / g(x)]
where both f(x) and g(x) approach 0 as x approaches c or both f(x) and g(x) approach infinity as x approaches c, and the limit exists, then you can differentiate the numerator and the denominator and take the limit of the resulting quotient:
Lim (x → c) [f'(x) / g'(x)]
Here, f'(x) and g'(x) represent the derivatives of the functions f(x) and g(x).
L'Hôpital's Rule is a powerful tool for evaluating limits in cases where direct substitution would result in an indeterminate form. However, it's essential to verify that the conditions for using the rule are met, and it's not always applicable to all types of limits. If the limit of the quotient of derivatives still results in an indeterminate form, you may need to apply L'Hôpital's Rule repeatedly until the limit converges or find an alternative approach to evaluate the limit.
Solved Examples on L'Hopital's Rule
L'Hôpital's Rule is a mathematical technique used to evaluate limits of indeterminate forms, particularly when you have a fraction where both the numerator and denominator tend to zero or infinity. Here are some solved examples to illustrate how L'Hôpital's Rule works:
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Example 1: Find the limit as x approaches 0 of (sin(x) / x).
Solution:
- Take the derivative of the numerator: d/dx(sin(x)) = cos(x).
- Take the derivative of the denominator: d/dx(x) = 1.
So, lim(x→0) (sin(x) / x) = 1.
Example 2: Find the limit as x approaches infinity of (e^x / x).
Solution:
- Take the derivative of the numerator: d/dx(e^x) = e^x.
- Take the derivative of the denominator: d/dx(x) = 1.
So, lim(x→∞) (e^x / x) = ∞.
Example 3: Find the limit as x approaches 0 of (x^2 / e^x).
Solution:
- Take the derivative of the numerator: d/dx(x^2) = 2x.
- Take the derivative of the denominator: d/dx(e^x) = e^x.
- Take the derivative of the numerator: d/dx(2x) = 2.
- Take the derivative of the denominator: d/dx(e^x) = e^x.
So, lim(x→0) (x^2 / e^x) = 2.
These examples demonstrate how L'Hôpital's Rule can be used to evaluate limits in cases where direct substitution leads to indeterminate forms. It's important to note that L'Hôpital's Rule is applicable when the limit has the form 0/0 or ∞/∞, and it can be applied repeatedly as needed.
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