Question

In L'Hôpital's 1696 calculus textbook, he illustrated his rule using the limit of the function \(f(x)=\frac{\sqrt{2 a^{3} x-x^{4}}-a \sqrt[3]{a^{2} x}}{a-\sqrt[4]{a x^{3}}}\) as \(x\) approaches \(a, a>0 .\) Find this limit.

Short answer

The limit of the function as \(x\) approaches \(a\) is determined by successive applications of L'Hôpital's Rule until a determinate form is found. The exact value of the limit depends on the derivatives of the functions involved and cannot be determined without further mathematical computation.

Step-by-step solution

Recognize the Indeterminate Form

Observe whether the function \(f(x)\) forms an indeterminate form as \(x\) approaches \(a\). Set \(x=a\) in the function. Upon simplification, find that it results in \(\frac{0}{0}\), an indeterminate form.

Apply L'Hôpital's Rule

For this indeterminate form, apply L'Hôpital's Rule, which states that the limit of ratios of two functions as x approaches a value might be found by taking the limit of the derivatives of the functions. Therefore, differentiate the numerator and the denominator of the function with respect to \(x\).

Compute and Simplify the Derivatives

Compute the derivative of both the numerator and the denominator using the chain rule, product rule and the power rule of derivatives. Subsequent simplification may involve several algebraic steps.

Determine the Limit using the Derivatives

Substitute \(x=a\) into the derivative of the numerator and the derivative of the denominator. If the obtained expression is a determinate form, then this is the limit; if not, reapply L'Hôpital's Rule.

Repeat Application of L'Hôpital's Rule if Necessary

If the resulting expression after substitution still results in an indeterminate form, reapply L'Hôpital’s Rule. Find the derivatives of the earlier derived numerator and denominator expressions. Repeat the process until a determinate form is obtained. After finding the limit, check if the result is valid.