Introduction

Mathematics is a fundamental tool in understanding and solving complex problems across various fields, including science, engineering, and business. In this article, we will delve into the calculation of a specific limit, focusing on the value of the following expression:

Understanding the Mathematical Expression

The mathematical expression we are interested in is:

(lim_{x to 0} frac{x4^{frac{3}{2}}e^x - 9}{x})

This expression represents a complex limit that needs to be evaluated. To solve this, we will apply the principles of calculus and L'H?pital's Rule.

Evaluating the Limit Using L'H?pital's Rule

L'H?pital's Rule is a well-known theorem that is used to evaluate indeterminate forms, particularly when the limit of a quotient of functions is of the form 0/0 or ∞/∞. In our case, the expression simplifies to a form suitable for applying L'H?pital's Rule.

Step 1: Initial Limit Calculation

First, let's rewrite the given expression:

[lim_{x to 0} frac{x4^{frac{3}{2}}e^x - 9}{x}]

Simplifying the numerator, we get:

[lim_{x to 0} frac{1}{x}left[81 cdot x cdot 4^{frac{3}{2}}e^x - 9right]]

Step 2: Applying L'H?pital's Rule

Given that the form is now more complex, let's apply L'H?pital's Rule. We differentiate the numerator and the denominator with respect to x:

Numerator: [81 cdot 4^{frac{3}{2}}e^x 81 cdot x cdot 4^{frac{3}{2}}e^x]

Denominator: (1)

Therefore, the expression simplifies to:

[lim_{x to 0} 81 cdot 4^{frac{3}{2}}e^x 81 cdot x cdot 4^{frac{3}{2}}e^x]

Step 3: Simplifying the Expression

Evaluating at x 0, we get:

[81 cdot 4^{frac{3}{2}}e^0 81 cdot 8 cdot 1 648]

Considering the application of L'H?pital's Rule, the simplified form of the limit is:

[lim_{x to 0} left[81 cdot 4^{frac{3}{2}}e^x 81 cdot x cdot 4^{frac{3}{2}}e^xright] 648]

Understanding the Simplified Form

Upon further simplification, we realize that the terms involving x converge to 0, leaving us with:

[lim_{x to 0} 81 cdot 4^{frac{3}{2}}e^0 648]

Conclusion

Through the application of L'H?pital's Rule, we have determined that the limit of the complex mathematical expression is 648. This method provides a clear and structured approach to solving such problems, highlighting the importance of calculus and its applications in various scientific and mathematical contexts.