Question

In Exercises \(33-38,\) use a computer algebra system to graph the function and find \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) (if it exists). \(f(x, y)=\sin x+\sin y\)

Short answer

The limit of the function \( f(x, y) = \sin x + \sin y \) as \((x, y)\) approaches \((0,0)\) is 0.

Step-by-step solution

Graphing the function

Using a computer algebra system, graph the function \( f(x, y) = \sin x + \sin y \). This step will give us a visual representation of the function's behavior in space.

Finding the limit

To find \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\), you can apply the limit to \( x \) and \( y \) separately, since they are independent variables. The limit of \( \sin x \) as \( x \) approaches \( 0 \) is \( 0 \), and similarly for \( y \). So, \(\lim _{(x, y) \rightarrow(0,0)} f(x, y) = 0 + 0 = 0\).

Key concepts

Computer Algebra System

A Computer Algebra System (CAS) is an essential tool in advanced mathematics, particularly in handling complex algebraic structures and computations. CAS software allows us to perform symbolic mathematics, which means it can manipulate algebraic expressions directly rather than simply performing numerical calculations.

For tasks like graphing multivariable functions and finding limits, a CAS is invaluable. It understands commands for differentiation, integration, and, in our context, graphing functions like f(x, y) = sin x + sin y. By using a CAS, you can visualize how the function behaves in a two-dimensional or three-dimensional space without doing the plotting by hand.

Moreover, when it comes to limits, a CAS can either provide a direct answer or assist in simplifying the problem to a solvable state, especially when dealing with multivariable limits. This computational power not only saves time but often reveals details about the overall structure and patterns within the mathematical problem that might not be visible otherwise.

Graphing Functions

Graphing is an essential tool in calculus, offering a graphical representation that helps in understanding the behavior of functions. When graphing functions, especially those with two or more variables, we obtain a visual landscape which can give immediate insights into the function's properties, such as continuity, extrema, and the areas where the function increases or decreases.

In multivariable calculus, graphing is more complex than in single-variable calculus because you are dealing with surfaces or contours rather than simple curves. For the function f(x, y) = sin x + sin y, graphing helps us visualize how this function behaves as both x and y approach zero, which aids in confirming the results we might get analytically when evaluating the limit.

To make the most of graphing, understanding the axes and scales is crucial since mistakes here can lead to misinterpretation of the function's behavior. Properly graphed, the function provides visual confirmation and can be a strong companion to the analytical methods used for finding limits.

Limits in Calculus

The concept of limits is foundational in calculus, as it helps describe the behavior of functions as inputs approach a particular point. When discussing limits in calculus, we often focus on a function approaching a certain x-value, but in multivariable calculus, we investigate how a function behaves as it approaches a point in space, such as ewline(0,0).

In the given exercise, the multivariable limit ewline ewline(0,0)} f(x, y)ewline involves approaching the point (0,0) from any direction in the xy-plane. Methods to find this limit include direct substitution (if applicable), factoring and simplifying, or using special techniques like the squeeze theorem, polar coordinates, or paths.

Finding multivariable limits can be tricky because you have to ensure that the limit is the same, regardless of the path taken to approach the point. If different paths lead to different limits, the overall limit does not exist. For the function in our exercise, a direct substitution shows us that as x and y separately approach zero, the function approaches zero as well, confirming that ewline(0,0)} f(x, y) = 0ewline. This step-by-step reasoning is crucial in grasping the limit concept fully.