91Ó°ÊÓ

Multivariable calculus deals with functions involving more than one variable. In our exercise, the function \( f(x,y) = e^{\sqrt{4x^2 + 3y^2}} \) includes two variables: \( x \) and \( y \). This makes the process of finding limits slightly more complex. We do not only consider one direction of approach, like you would with single-variable limits. Instead, the point \((x, y) \rightarrow (1, 3)\) can be approached from infinitely many paths in the plane.

To solve problems like these, it’s important to confirm that the limit approaches the same value regardless of the path. This might involve using techniques like paths along lines, curves, or direct substitution when possible. Doing so helps to ensure the limit exists and is unique. In this exercise, direct substitution was sufficient because the function is continuous at the point of interest.

Understanding the behavior of the function \( f(x,y) = e^{\sqrt{4x^2 + 3y^2}} \) is key. As \((x, y)\) approaches \((1, 3)\), the expression inside the square root, \(4x^2 + 3y^2\), becomes \(4(1)^2 + 3(3)^2 = 31\).

The function composition layers should be analyzed:

• **Square root behavior:** Inside the square root, \(4x^2 + 3y^2\) transitions from variables to the constant 31. The square root of a positive number is nicely defined, causing no interruptions or undefined behavior in the limit check.
• **Exponential function:** Exponential functions \(e^x\) are continuous and smooth. As such, applying the exponential to the square root result makes the full function also well-behaved.

This layered approach ensures that \( f(x, y) \) behaves consistently as \((x, y)\) nears \((1, 3)\).

Exponential functions, like \( e^x \), play an essential role in calculus due to their properties and continuous growth. When dealing with \( e^{\sqrt{31}} \), this function maintains continuity and differentiability over its entire domain.

Exponential functions are characterized by:

• **Rapid growth:** As the exponent increases, \( e^x \) grows quickly. This behavior is crucial when determining limits, as any discontinuities or undefined values inside the exponent can impact the limit outcome.
• **Always positive:** \( e^x \) is always positive, which helps avoid potential pitfalls around zero or negative values in limit calculus.

In this exercise, once the inner operation evaluated to \(\sqrt{31}\), the resulting exponential function, \(e^{\sqrt{31}}\), gives us a precise limit value.