Question

Evaluate one of the limits I'Hôpital used in his own textbook in about 1700: \(\lim _{x \rightarrow a} \frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}}{a-\sqrt[4]{a x^{3}}},\) where \(a\) is a real number.

Short answer

Question: Evaluate the limit of the given expression $\lim_{x\to a} \frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}}{a-\sqrt[4]{a x^{3}}}$ Answer: The limit as $x$ approaches $a$ of the given expression is $\frac{8a-3a^{2/3}}{-9}$

Step-by-step solution

Find the derivatives of the numerator and the denominator.

To find the derivative of the numerator, use the chain rule: $$\frac{d}{dx}\left(\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}\right)$$ $$= \frac{1}{2\sqrt{2a^3x-x^4}}(6a^2x-4x^3)-\frac{a^2}{3\sqrt[3]{(a^2x)^2}}$$ Next, find the derivative of the denominator: $$\frac{d}{dx}\left(a-\sqrt[4]{a x^{3}}\right)$$ $$= -\frac{3}{4}a\sqrt[4]{x^2}$$

Apply L'Hôpital's Rule.

Now that we have found the derivatives, we can apply L'Hôpital's Rule, provided the limit of the ratio of their derivatives exists. This means we need to find: $$\lim_{x\to a} \frac{\frac{1}{2\sqrt{2a^3x-x^4}}(6a^2x-4x^3)-\frac{a^2}{3\sqrt[3]{(a^2x)^2}}}{-\frac{3}{4}a\sqrt[4]{x^2}}$$

Simplify the expression and find the limit as x approaches a.

Simplify the expression by canceling terms: $$\lim_{x\to a} \frac{\frac{6a^2x-4x^3}{2\sqrt{2a^3x-x^4}}-\frac{a^2}{3\sqrt[3]{(a^2x)^2}}}{-\frac{3}{4}a\sqrt[4]{x^2}}$$ $$= \lim_{x\to a} \frac{4(6a^2x-4x^3)-3a^2(2\sqrt{2a^3x-x^4})^{1/3}}{-9a(2a^3x-x^4)^{1/4}x}$$ Now, find the limit as x approaches a: $$\frac{4(6a^3-4a^3)-3a^2(2a^4-a^4)^{1/3}}{-9a(2a^4-a^4)^{1/4}a}$$ $$= \frac{4(2a^3)-3a^2a^{4/3}}{-9a^2}$$ $$= \frac{8a^3-3a^{8/3}}{-9a^2}$$ Which simplifies to: $$\frac{8a-3a^{2/3}}{-9}$$ So the limit is: $$\lim_{x\to a} \frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}}{a-\sqrt[4]{a x^{3}}} = \frac{8a-3a^{2/3}}{-9}$$

Key concepts

Understanding the Chain Rule

The chain rule is a fundamental tool in calculus for finding the derivative of a composite function. It helps us differentiate functions nested within each other. If you have a function defined as \( f(g(x)) \), the chain rule states that its derivative is \( f'(g(x)) \cdot g'(x) \). Simply put, you differentiate the outer function and multiply by the derivative of the inner function.

Using the chain rule is essential when dealing with complex functions, such as nested square roots or cube roots. It's applied to functions in our original exercise, where we needed to differentiate expressions like \( \sqrt{2 a^3 x - x^4} \) and \( \sqrt[3]{a^2 x} \).

Key steps in using the chain rule effectively are:

• Identify the outer and inner functions.
• Differentiate both functions separately.
• Multiply the derivative of the outer function by the derivative of the inner function.

This technique gets you through intricate differentiation processes found in many calculus problems. It is one of the core methods used to simplify the expressions in L'Hôpital's Rule applications.

Approaching Limits

Limits are central to calculus, acting as the foundation for derivatives and integrals. They describe the behavior of a function as it approaches a particular point, helping to explore functions' behavior at points that might not be directly accessible, such as where a function becomes undefined.

In our exercise, we aimed to find the limit of a complex expression using L'Hôpital's Rule, which is helpful when we have indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). This rule allows us to substitute the limit of a complicated expression with the limit of its derivatives, under specific conditions.

Steps involved in evaluating limits using L'Hôpital's Rule are:

• Confirm the limit leads to an indeterminate form.
• Differentiate the numerator and denominator separately.
• Find the limit of the resulting derivative expressions as \( x \) approaches the given point.

Limits provide deeper insight into the continuity and behavior of functions and are critical in evaluating real-world phenomena where direct calculation is not feasible.

Differentiating Derivatives

Derivatives are a measure of how a function changes as its input changes, providing the rate of change at any given point. In the exercise, we found derivatives of a complex numerator and denominator to simplify and solve the limit problem using L'Hôpital's Rule.

Derivative concepts like power rule, chain rule, and derivative of a root function came into play. Calculating derivatives helps in understanding behavior such as slope, velocity, acceleration, and other rate-of-change quantities in various fields.

Key derivatives from the exercise:

• The derivative of \( \sqrt{2 a^3 x - x^4} \) utilizes the chain rule for differentiation.
• The derivative of \( a\sqrt[3]{a^2 x} \) incorporates the chain rule to handle the cube root.
• The derivative of the denominator employs the chain rule for a fourth root expression.

Mastering derivatives equips you with the tools needed for solving complex calculus problems and serves as a stepping stone for advanced mathematical concepts.