/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 58 Evaluate the following limits. ... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 58

Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1,-1,1)} \frac{x z+5 x+y z+5 y}{x+y}$$

Short Answer

Expert verified
$$ Answer: The limit does not exist.

Step by step solution

01

Plug in the limit point and simplify the numerator

To do this, substitute \(x=1\), \(y=-1\), \(z=1\) into the given expression: $$\frac{(1)(1)+5(1)+(-1)(1)+5(-1)}{1-1}$$
02

Simplify the expression

Simplify the expression by combining like terms in the numerator and denominator: $$\frac{1+5-1-5}{0}$$ Simplify the fraction: $$\frac{0}{0}$$ Since we have an indeterminate form \(\frac{0}{0}\), we will need to explore other techniques to find the limit.
03

Apply L'Hospital's Rule (multivariate version)

Unfortunately, L'Hospital's Rule is not directly applicable in this multivariate case. Instead, let's find a different path and use polar coordinates for solving this limit issue.
04

Switch to polar coordinates

Replace \((x, y, z)\) with polar coordinates \((r, \theta, \phi)\), where: $$x = 1 + r\cos{\theta}\sin{\phi}$$ $$y = -1 + r\sin{\theta}\sin{\phi}$$ $$z = 1 + r\cos{\phi}$$ As \((x, y, z) \rightarrow(1,-1,1)\), we have \((r, \theta, \phi) \rightarrow(0, \theta, \phi)\).
05

Substitute the polar coordinates into the expression

Replace \(x\), \(y\), and \(z\) by their polar equivalents: $$\frac{(1+r\cos\theta\sin\phi)(1+r\cos\phi)+5(1+r\cos\theta\sin\phi)+(-1+r\sin\theta\sin\phi)(1+r\cos\phi)+5(-1+r\sin\theta\sin\phi)}{(1+r\cos\theta\sin\phi)+(-1+r\sin\theta\sin\phi)}$$
06

Evaluate the limit as \(r \rightarrow 0\)

We can now try to find the limit as \(r \rightarrow 0\): $$\lim _{r \rightarrow 0} \frac{(1+r\cos\theta\sin\phi)(1+r\cos\phi)+5(1+r\cos\theta\sin\phi)+(-1+r\sin\theta\sin\phi)(1+r\cos\phi)+5(-1+r\sin\theta\sin\phi)}{(1+r\cos\theta\sin\phi)+(-1+r\sin\theta\sin\phi)}$$ Notice that all terms containing \(r\) in both the numerator and the denominator will go to zero as \(r\rightarrow 0\), leaving only the constants. Thus, the expression becomes: $$\frac{(1)(1)+5(1)+(-1)(1)+5(-1)}{(1)+(-1)}$$ Which is exactly the same expression as in Step 2. Unfortunately, we could not avoid the indeterminate form \(\frac{0}{0}\). Therefore, the provided limit does not exist.

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Most popular questions from this chapter

Problems with two constraints Given a differentiable function \(w=f(x, y, z),\) the goal is to find its maximum and minimum values subject to the constraints \(g(x, y, z)=0\) and \(h(x, y, z)=0\) where \(g\) and \(h\) are also differentiable. a. Imagine a level surface of the function \(f\) and the constraint surfaces \(g(x, y, z)=0\) and \(h(x, y, z)=0 .\) Note that \(g\) and \(h\) intersect (in general) in a curve \(C\) on which maximum and minimum values of \(f\) must be found. Explain why \(\nabla g\) and \(\nabla h\) are orthogonal to their respective surfaces. b. Explain why \(\nabla f\) lies in the plane formed by \(\nabla g\) and \(\nabla h\) at a point of \(C\) where \(f\) has a maximum or minimum value. c. Explain why part (b) implies that \(\nabla f=\lambda \nabla g+\mu \nabla h\) at a point of \(C\) where \(f\) has a maximum or minimum value, where \(\lambda\) and \(\mu\) (the Lagrange multipliers) are real numbers. d. Conclude from part (c) that the equations that must be solved for maximum or minimum values of \(f\) subject to two constraints are \(\nabla f=\lambda \nabla g+\mu \nabla h, g(x, y, z)=0,\) and \(h(x, y, z)=0\)

The domain of $$f(x, y)=e^{-1 /\left(x^{2}+y^{2}\right)}$$ excludes \((0,0) .\) How should \(f\) be defined at (0,0) to make it continuous there?

Identify and briefly describe the surfaces defined by the following equations. $$x^{2} / 4+y^{2}-2 x-10 y-z^{2}+41=0$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Suppose you are standing at the center of a sphere looking at a point \(P\) on the surface of the sphere. Your line of sight to \(P\) is orthogonal to the plane tangent to the sphere at \(P\). b. At a point that maximizes \(f\) on the curve \(g(x, y)=0,\) the dot product \(\nabla f \cdot \nabla g\) is zero.

Use the formal definition of a limit to prove that $$\lim _{(x, y) \rightarrow(a, b)} c f(x, y)=c \lim _{(x, y) \rightarrow(a, b)} f(x, y)$$
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