Problem 70 - Calculus Volume 3

Question

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(4,4)} x \ln y$$

Step-by-Step Solution

Answer

The limit is approximately 5.5452.

Step 1: Substitute Values

To evaluate the limit \( \lim _{(x, y) \rightarrow(4,4)} x \ln y \), we begin by substituting the values \(x = 4\) and \(y = 4\) directly into the function \(x \ln y\). This gives us \(4 \ln 4\).

Step 2: Evaluate Natural Logarithm

Next, we need to calculate \(\ln 4\). Since the natural logarithm is a standard function, we can use a calculator to find that \(\ln 4 \approx 1.3863\).

Step 3: Multiply

Finally, multiply the value of \(\ln 4\) by \(4\) to evaluate the limit: \[ 4 \times 1.3863 \approx 5.5452. \]

Step 4: Conclusion: State the Limit

Thus, the limit \( \lim _{(x, y) \rightarrow(4,4)} x \ln y = 5.5452\).

Key Concepts

Limits of Functions

In multivariable calculus, the concept of limits extends to functions of multiple variables. A limit tries to determine the behavior of a function as the variables approach certain values. It’s slightly more complex than simple single-variable limits. When we analyze the limit \[ \lim _{(x, y) \rightarrow(4,4)} x \ln y \]we are assessing what the output of the function \(x \ln y \) will be as both \(x\) and \(y\) head towards 4.**Key points to remember**:

• Make sure the values approach the same point from all directions. If they don’t and the limit varies, it may not exist.
• Understand how the function behaves around those points without actually reaching them.

Limits are foundational in understanding continuity, evaluating derivatives, and integrating functions. They’re essential in solving real-world problems, particularly in physics and engineering. Ultimately, knowing how to effectively work with limits of multivariable functions greatly enhances your mathematical toolkit.

Natural Logarithm

The natural logarithm, denoted as \(\ln\),is a logarithm with the base \(e\),where \(e\approx2.71828\),which is Euler's number. It's an essential concept in calculus as it helps to describe growth processes, much like interest compounding on a bank account.**Exploring Natural Logarithms**:

• Natural logarithms are preferred in calculus due to their simple differentiation and integration properties.
• They show up frequently in real-life scenarios associated with exponential growth and decay, such as population growth or radioactive decay.

When evaluating the expression\(x \ln y\), the role of the natural logarithm is to add complexity to the function by associating continuous growth to the variable \(y\).The natural logarithm of 4,which is approximately 1.3863,gives insight into the functional output beyond linear scaling.

Evaluating Limits

Evaluating limits, especially in multivariable calculus, requires careful substitution and analysis. The aim is to find what value a function approaches as the variables get infinitely close to certain points. **Step-by-step approach**:

• Start by substituting values directly into the function, if possible. Here, you substitute specific values such as \(x = 4\) and \(y = 4\) in \(x \ln y\).
• Handle any algebraic or logarithmic simplifications. Calculate any constants like \(\ln 4\).
• Finish by doing the necessary arithmetic to reach the final value, which in our case, arrives at approximately 5.5452.

Avoid common pitfalls by ensuring there is no division by zero or disagreement in limit from different paths. Evaluating limits smoothly takes practice but becomes invaluable across all fields of calculus.