Question

Verify that \(\lim _{(x, y) \rightarrow(0,0)} \frac{\sin x+\sin y}{x+y}=1.\)

Short answer

Question: Verify that the limit $$\lim _{(x, y) \rightarrow(0,0)} \frac{\sin x+\sin y}{x+y}=1.$$ Answer: By using the properties of limits and trigonometric identities, we have shown that the given limit is equal to 1, as shown in the solution above.

Step-by-step solution

Write down the limit

We need to verify that $$\lim _{(x, y) \rightarrow(0,0)} \frac{\sin x+\sin y}{x+y}=1.$$

Apply trigonometric identity

We can rewrite the numerator using the sum to product identity. \(\sin(x) + \sin(y) = 2 \sin(\frac{x+y}{2})\cos(\frac{x-y}{2})\). So the expression becomes: $$\frac{2 \sin(\frac{x+y}{2})\cos(\frac{x-y}{2})}{x+y}$$

Divide the expression into two fractions

Next, let's split the expression into two fractions: $$\frac{2 \sin(\frac{x+y}{2})}{x+y} \cdot \cos(\frac{x-y}{2})$$

Apply the limit

Now, find the limit: $$\lim_{(x, y) \rightarrow (0,0)} \frac{2 \sin(\frac{x+y}{2})}{x+y} \cdot \cos(\frac{x-y}{2})$$

Apply properties of limits

Using the properties of limits, we can split the given limit and find the product of individual limits: $$\lim_{(x, y) \rightarrow (0,0)} \frac{2 \sin(\frac{x+y}{2})}{x+y} \cdot \lim_{(x, y) \rightarrow (0,0)} \cos(\frac{x-y}{2})$$

Evaluate the individual limits

The first limit is a well-known limit from single variable calculus: $$\lim_{z \rightarrow 0} \frac{\sin(z)}{z} = 1$$ Substitute \(z = \frac{x+y}{2}\) and we get: $$\lim_{\frac{x+y}{2} \rightarrow 0} \frac{2 \sin(\frac{x+y}{2})}{x+y} = 1$$ As for the second limit, since cosine of any finite value is bounded, we have: $$\lim_{(x, y) \rightarrow (0,0)} \cos(\frac{x-y}{2}) = \cos(0) = 1$$

Combine the limits

Now, combining the results of individual limits: $$\lim_{(x, y) \rightarrow (0,0)} \frac{\sin x+\sin y}{x+y} = 1 \cdot 1 = 1$$ This verifies the given limit.

Key concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that express relationships between trigonometric functions. They play a key role in simplifying and solving problems involving angles and distances in geometry and calculus. For our exercise, the relevant identity is the sum to product identity: - \(\sin(x) + \sin(y) = 2 \sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)\) - This identity helps convert the sum of sines into a product, which often simplifies problems, especially those involving limits. Utilizing such identities bridges the gap between seemingly complex terms and manageable expressions. There are other essential trigonometric identities, including Pythagorean identities, and double angle identities, each serving a purpose in mathematics.

Limit Properties

Understanding the properties of limits is essential to evaluating expressions in calculus, particularly when dealing with multivariable scenarios. Limit properties, such as the product, quotient, and sum laws, enable the breaking down of complex problems into simpler, solvable parts. Key Limit Properties Include: - Limit of a constant multiple: \( \lim_{x\to a} c f(x) = c \lim_{x\to a} f(x)\) - Limit of a sum: \( \lim_{x\to a} [f(x) + g(x)] = \lim_{x\to a} f(x) + \lim_{x\to a} g(x)\) - Limit of a product: \( \lim_{x\to a} [f(x)g(x)] = \lim_{x\to a} f(x) \cdot \lim_{x\to a} g(x)\) In our exercise, the product rule helps break down the original expression into individual limits, making it easier to solve by evaluating each component separately.

Sum to Product Identity

The sum to product identities are a subset of trigonometric identities that are particularly useful for expressions involving addition or subtraction of trigonometric functions. The primary purpose is to convert sums into products, which can be easier to integrate, differentiate, or use in limit evaluations. For the exercise, we used the identity: - \(\sin(x) + \sin(y) = 2 \sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)\) This conversion transforms the original limit into a simpler form, showing its true value. Such identities are indispensable tools in both pure and applied mathematics, seen often in signals processing, physics, and engineering for solving complex systems.

Single Variable Calculus Limits

Single variable calculus limits allow us to determine the behavior of functions as they approach particular points. A fundamental limit that often emerges is \(\lim_{z \to 0} \frac{\sin(z)}{z} = 1\). This helps assess the limiting behavior of trigonometric functions, especially when dealing with approximation or small angle scenarios. In our multivariable problem, it transforms into: - \(\lim_{\frac{x+y}{2} \to 0} \frac{2 \sin(\frac{x+y}{2})}{x+y} = 1\) This limit is pivotal to evaluating the given expression as it reveals the form which approaches a known standard outcome. Grasping these single variable examples aids in building understanding of more complex, higher-dimensional scenarios, like multivariable calculus.