Problem 8 - Calculus with Concepts in Calculus

Question

Evaluate the limit. $$ \lim _{(x, y) \rightarrow(\ln 2,0)} e^{2 x+y^{2}} $$

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Understand the Expression

We need to evaluate the limit of the function \( f(x, y) = e^{2x + y^2} \) as \((x, y)\) approaches \((\ln 2, 0)\). Our task is to substitute these values into the expression and evaluate it.

Step 2: Substitute the Approaching Values

Substitute \( x = \ln 2 \) and \( y = 0 \) directly into the expression. This gives us \( e^{2(\ln 2) + 0^2} = e^{2(\ln 2)} \).

Step 3: Simplify the Expression

Recall the logarithmic identity \( a \ln b = \ln b^a \). Thus, \( 2 \ln 2 = \ln 2^2 = \ln 4 \). We can rewrite our expression as \( e^{\ln 4} \).

Step 4: Evaluate the Expression

Since the exponential and logarithm functions are inverses, \( e^{\ln 4} = 4 \). Therefore, the expression simplifies directly to 4.

Key Concepts

Exponential Function

The exponential function is a fundamental mathematical concept denoted by \( e^x \), where \( e \) is the base of natural logarithms, approximately equal to 2.71828. This type of function is unique because it grows rapidly and continuously. It is widely used in various fields such as biology, economics, and physics due to its properties of continuous growth and ability to model real-world phenomena like population growth or radioactive decay.

The hallmark of exponential functions is their constant relative growth rate: the derivative of \( e^x \) is \( e^x \), implying that the rate of change of the function is proportional to its current value. This self-similarity makes exponential functions powerful in both pure and applied mathematics.

In the context of multivariable calculus, expressions like \( e^{2x + y^2} \) involve exponential functions with multivariable inputs. This means that the exponential growth is influenced by more than one variable at a time, which can make analysis more complex but also more representative of real-world systems.

Logarithmic Identity

Logarithmic identities are crucial in simplifying mathematical expressions, particularly when dealing with limits involving exponentials and logarithms. One vital identity is \( a \ln b = \ln b^a \). This property allows the simplification of exponential functions when combined with logarithms.

In our problem, we use this identity to simplify \( 2 \ln 2 \) into \( \ln 4 \). This transformation is key to evaluating the limit efficiently. Understanding how logarithmic identities work is essential, especially in calculus where simplifying expressions can turn complex problems into easier ones to solve.

These identities are not just mathematical tricks; they showcase how the exponential and logarithmic functions are inverses, allowing them to "undo" each other. Recognizing these patterns is invaluable for tackling calculus problems, particularly when you encounter tasks that require transforming expressions astutely.

Multivariable Calculus

Multivariable calculus extends calculus from functions of a single variable to functions of several variables. This branch of mathematics is crucial for understanding and navigating spaces of higher dimensions.

When working with multivariable functions like \( f(x, y) = e^{2x + y^2} \), we are interested in how the function behaves as it approaches particular points in a plane or space. Instead of moving along a line, the input values approach from various paths, adding complexity to limit evaluation.

In evaluating the limit \( \lim_{(x, y) \rightarrow(\ln 2,0)} e^{2x+y^{2}} \), we examine the behavior of \( f(x, y) \) as \( x \) approaches \( \ln 2 \) and \( y \) approaches 0. Understanding how to manage multiple approaches and simplify expressions like \( e^{\ln 4} \) is a testament to mastering multivariable calculus tools. These skills are fundamental in many scientific and engineering fields, where precise analysis of systems with several interdependent variables is necessary.