Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\lim _{(x, y) \rightarrow(0,0)} \frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}\)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the target limit

We are tasked with finding the limit of the function \( f(x, y) = \frac{\sin(x^{2}+y^{2})}{x^{2}+y^{2}} \) as \( (x, y) \to (0,0) \).

Step 2: Convert to polar coordinates

In polar coordinates, \( x^{2} + y^{2} = r^{2} \), where \( r \) is the radius from the origin. So, the function becomes \( \frac{\sin(r^{2})}{r^{2}} \).

Step 3: Apply known limit results

Recall the standard limit \( \lim_{r \to 0} \frac{\sin(r)}{r} = 1 \). This can be applied to our function by setting \( u = r^{2} \), which gives \( \lim_{r \to 0} \frac{\sin(r^{2})}{r^{2}} = \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \).

Step 4: Conclude the limit value

As \( r \rightarrow 0 \), \( \frac{\sin(r^{2})}{r^{2}} \rightarrow 1 \). Therefore, the original limit \( \lim _{(x, y) \rightarrow(0,0)} \frac{\sin(x^{2}+y^{2})}{x^{2}+y^{2}} = 1 \).

Key Concepts

Polar Coordinates

Polar coordinates are an alternative way of representing points in a plane, especially useful for systems involving circular motion or symmetry around a point. Instead of using Cartesian coordinates

• A point is defined by a distance from a fixed point (the origin), known as the radius (\( r \)).
• An angle from a fixed direction, often the positive x-axis, called the polar angle (\( \theta \)).

For example, if a point has coordinates (\( x, y \)), it can be expressed in polar coordinates as (\( r \), \( \theta \)) where \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

In our multivariable calculus limit problem, converting \( x^2 + y^2 \) into \( r^2 \) simplifies the problem into a form that uses the symmetry of the circle. This makes it easier to work with functions and limits when \( r \to 0 \), focusing on the distance to the origin without worrying about the particular path.

Limits in Two Variables

Limits in functions of two variables \( f(x, y) \) explore the behavior of \( f \) as \( (x, y) \) approaches some point, like \( (0, 0) \). These limits can be more complex than their single-variable counterparts due to multiple paths to approach the limit point.

Approaching \( (0,0) \) can be done along various paths, such as the x-axis, y-axis, or diagonals, leading to potentially different function values unless handled properly. Therefore, confirming a limit through different paths or using polar coordinates to reduce the multiple-variable problem to a single-variable form often helps in verifying consistency.

• The polar conversion changes the variables into one parameter (\( r \)), making it easier to see that at \( r = 0 \), the resultant value is consistent no matter the direction of approach.

In our exercise, by using polar coordinates, the limit \( \lim _{(x, y) \rightarrow(0,0)} \frac{\sin(x^{2}+y^{2})}{x^{2}+y^{2}} \) simplifies to examining \( \lim_{r \to 0} \frac{\sin(r^{2})}{r^{2}} \), circumventing the trouble of evaluating it separately along different paths.

Trigonometric Limits

The exploration of trigonometric limits is a fundamental topic in calculus, deeply tied to understanding the behavior of functions involving trigonometric expressions. A classic result often used in calculus is the limit\( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \). This limit showcases how the sine function behaves around its origin and is pivotal for addressing many calculus problems involving sine.

In our specific limit problem, the formula \( \frac{\sin(u)}{u} \) serves as a critical tool for evaluating \( \frac{\sin(r^{2})}{r^{2}} \)as \( r \to 0 \). By setting \( u = r^2 \), we directly apply this standard limit to solve our multivariable problem. This reduces the complexity of evaluating the original expression, essentially shifting a two-variable problem to a familiar single-variable trigonometric limit.

• This technique highlights the utility of converting a complex expression into simpler known limits, effectively employing core trigonometric relationships to find required values.