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Chapter 4: Problem 69

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. $$ \lim _{(x, y) \rightarrow(2,5)}\left(\frac{1}{x}-\frac{5}{y}\right) $$

Short Answer

Expert verified
The limit is \(-\frac{1}{2}\).

Step by step solution

01

Identify the Limit Expression

The given limit expression is \( \lim _{(x, y) \rightarrow(2,5)}\left(\frac{1}{x}-\frac{5}{y}\right) \). We need to find the value of this expression as \(x\) approaches 2 and \(y\) approaches 5.
02

Substitute the Values of \(x\) and \(y\)

To find the limit as \((x, y)\) approaches \((2, 5)\), we substitute \(x = 2\) and \(y = 5\) into the expression: \( \frac{1}{x} - \frac{5}{y} \rightarrow \frac{1}{2} - \frac{5}{5} \).
03

Simplify the Expression

Simplify the substituted expression: \( \frac{1}{2} - \frac{5}{5} = \frac{1}{2} - 1 \). This simplifies to \( \frac{1}{2} - \frac{5}{5} = \frac{1}{2} - \frac{5}{5} = \frac{1}{2} - 1 = -\frac{1}{2} \).
04

State the Limit

Therefore, the limit of the expression as \((x, y)\) approaches \((2, 5)\) is \(-\frac{1}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit Evaluation
Limit evaluation is a fundamental concept in calculus that involves finding the value that a function approaches as the input gets closer to a specified point. For multivariable functions, like the one in our exercise, this means evaluating how the output behaves as both variables move towards specific values.
To evaluate the limit, you generally substitute the values of the variables into the expression and simplify. It becomes crucial to ensure the function is well-behaved around these values, which means no undefined operations, like division by zero.
  • For our exercise, we started by identifying the given limit expression, which is \( \lim _{(x, y) \rightarrow(2,5)}(\frac{1}{x}-\frac{5}{y}) \).
  • The next step was substituting the values directly, which is possible when the function is continuous and defined at those points.
  • The final step is simplifying to find the actual numerical limit, in this case, \( -\frac{1}{2} \).
This method works well when the observations can directly lead us to conclude about the approaching behavior at that point.
Multivariable Calculus
Multivariable calculus extends the concepts of single-variable calculus to more than one variable. This branch of mathematics is important for functions involving two or more variables. It evaluates how these functions behave when the input variables simultaneously change.
In the context of limits, multivariable calculus includes limits where one considers paths of approach. Unlike single-variable scenarios where there is only one direction, here, multiple paths can affect how a limit is approached.
  • Concepts like partial derivatives, gradients, and the notion of surfaces and curves in three-dimensional space stem from this branch.
  • Our problem demonstrates a multivariable situation where both \( x \) and \( y \) independently approach their respective values.
  • This significant aspect is ascertaining if different paths to the same point can yield different limit results or ensure continuity of the function.
Understanding this allows you to appreciate how input changes in a multi-dimensional sense, leading to real-world problem-solving in fields like engineering and physics.
Limit at a Point
A limit at a point involves calculating what value a function approaches as the input nears, but does not necessarily reach, specific coordinates. When dealing with a two-variable function like in our exercise, this means approaching from any direction within the plane.
The challenge is ensuring that the value the function approaches is independent of the path taken. For well-behaved functions, you evaluate the limit by direct substitution, as we did in our step-by-step solution.
  • We examined \( \lim _{(x, y) \rightarrow(2,5)}(\frac{1}{x}-\frac{5}{y}) \), and the approach was straightforward through points \( (2, 5) \).
  • This site-specific analysis requires ensuring no discontinuities or indeterminate forms obstruct the path.
  • It鈥檚 important in multiple dimensions to prove that regardless of the direction of approach, the limit value remains the same.
Successful computation reflects a deeper understanding of the function's continuity and behavior, establishing the reliability of function evaluations in practical scenarios.

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Most popular questions from this chapter

For the following exercises, use this information: A function \(f(x, y)\) is said to be homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) For all homogeneous functions of degree \(n, \quad\) the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\) . Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\) $$ f(x, y)=x^{2} y-2 y^{3} $$

For the following exercises, use this information: A function \(f(x, y)\) is said to be homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) For all homogeneous functions of degree \(n, \quad\) the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\) . Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\) $$ f(x, y)=\sqrt{x^{2}+y^{2}} $$

For the following exercises, find the derivative of the function. \(f(x, y, z)=\ln (x y+y z+z x) \) at point \((-9,-18,-27)\) in the direction the function increases most rapidly

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