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Chapter 2: Problem 7

For Exercises 7 -18, evaluate the given limit. $$ \lim _{(x, y) \rightarrow(0,0)} \cos (x y) $$

Short Answer

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Step by step solution

01

- Understanding the limit

The problem asks for the limit of the function \(\cos(xy)\) as the point \((x, y)\) approaches \(0,0\).
02

- Evaluate \(\cos(x y)\) at \((0,0)\)

To find the limit, we first evaluate \(\cos(x y)\) at the point \((0,0)\):\(\cos(0 \cdot 0) = \cos(0) = 1\).
03

- Analyze the function behavior

Consider the behavior of \(\cos(x y)\) as \((x,y)\) approaches \((0,0)\) from different paths. Since \(\cos(xy)\) is continuous everywhere and \(xy \rightarrow 0\) as \((x, y) \rightarrow (0, 0)\), the \(\cos(x y)\) will continuously approach \(1\).
04

- Confirm the limit

Since the function \(\cos(xy)\) does not depend on the path taken to approach \((0,0)\) and remains consistent, the limit is confirmed to be \(1\).

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Key Concepts

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

Multivariable Limits
Understanding multivariable limits is key when working with functions of two or more variables, such as \(\text{f(x, y)} \). A multivariable limit examines the behavior of a function as multiple variables approach specific values. For our exercise, we evaluate \(\text{f(x, y) = cos(xy)}\), as both \(\text{x}\) and \(\text{y}\) approach 0.
Continuous Functions
Continuous functions are those that do not have any breaks, jumps, or holes in their graph. They ensure the function's output changes smoothly without sudden interruptions. The cosine function, \(\text{cos(x)}\), is a perfect example of a continuous function. Because cosine is continuous, its limit at a point can be found by simply substituting the point into the function.
Cosine Function Behavior
The cosine function, \(\text{cos(胃)}\), varies between -1 and 1 for all real numbers \(\text{胃}\). As the product of \(\text{x}\) and \(\text{y}\) approaches 0, \(\text{cos(xy)}\) approaches \(\text{cos(0)}\), which equals 1. This continuous approach to a limit exemplifies why \(\text{cos(xy)}\) remains consistent, ultimately confirming that \(\text{lim \_{(x, y) \rightarrow (0,0)} cos(xy) = 1}\).

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Most popular questions from this chapter

Compute the gradient \(\nabla f\). $$ f(x, y)=x^{2}+y^{2}-1 $$

The function \(r(x, y)=\sqrt{x^{2}+y^{2}}\) is the length of the position vector \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}\) for each point \((x, y)\) in \(\mathbb{R}^{2}\). Show that \(\nabla r=\frac{1}{r} \mathbf{r}\) when \((x, y) \neq(0,0)\), and that \(\nabla\left(r^{2}\right)=2 \mathbf{r}\).

Use the substitution \(r=\sqrt{x^{2}+y^{2}}\) to show that $$ \lim _{(x, y) \rightarrow(0,0)} \frac{\sin \sqrt{x^{2}+y^{2}}}{\sqrt{x^{2}+y^{2}}}=1. $$ (Hint: You will need to use L'H么pital's Rule for single-variable limits.)

Find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$

Find the equation of the tangent plane to the surface \(z=f(x, y)\) at the point \(P\). \(f(x, y)=x^{2}+y^{3}, P=(1,1,2)\)
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