Problem 10 - Calculus Early Transcendentals

Question

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y \rightarrow(0,0)} \frac{x^{2}+\sin ^{2} y}{2 x^{2}+y^{2}}$$

Step-by-Step Solution

Answer

The limit does not exist as it depends on the path taken to approach zero.

Step 1: Check the Limit Along the x-axis

To find the limit, let's first approach the origin along the x-axis where \( y = 0 \). Substitute \( y = 0 \) into the expression:\[\lim_{{x \to 0}} \frac{x^{2} + \sin^{2}(0)}{2x^{2} + 0^{2}} = \lim_{{x \to 0}} \frac{x^{2}}{2x^{2}} = \lim_{{x \to 0}} \frac{1}{2} = \frac{1}{2}\]The limit along the x-axis is \( \frac{1}{2} \).

Step 2: Check the Limit Along the y-axis

Now, let's approach the origin along the y-axis where \( x = 0 \). Substitute \( x = 0 \) into the expression:\[\lim_{{y \to 0}} \frac{0^{2} + \sin^{2}(y)}{2(0^{2}) + y^{2}} = \lim_{{y \to 0}} \frac{\sin^{2}(y)}{y^{2}} \]Using the identity \( \sin(y) \approx y \) when \( y \to 0 \), this limit simplifies to \( \lim_{{y \to 0}} 1 = 1 \).The limit along the y-axis is \( 1 \).

Step 3: Compare the Calculated Limits

We found that the limit along the x-axis is \( \frac{1}{2} \) and along the y-axis is \( 1 \). Since the limits are not equal, the overall limit at \((0,0)\) does not exist.

Key Concepts

Two-variable functions

In multivariable calculus, we work with functions that have more than one input variable. These are known as two-variable functions. Specifically, they have the general form \(f(x, y)\). In this type of function, the output depends on two input variables, \(x\) and \(y\).
Two-variable functions create surfaces when plotted, providing a visual representation of how the outputs change with different inputs. For instance, the function used in our exercise is a two-variable function \(f(x, y) = \frac{x^{2}+ ext{sin}^{2}y}{2x^{2}+y^{2}}\).
Understanding two-variable functions is fundamental since it sets the stage for examining concepts like limits and continuity in multivariable calculus.

Limit along the x-axis

When finding limits for two-variable functions, one approach is to consider the limit along specific paths. The x-axis is a simple path where we set \(y = 0\).
In our exercise, after substituting \(y = 0\) into the function, it reduces to a single-variable function in terms of \(x\): \(\frac{x^{2}}{2x^{2}}\). As \(x\) approaches zero, we evaluate the limit:

• \(\lim_{x \to 0} \frac{1}{2} = \frac{1}{2}\)

This shows the behavior of the function as it approaches the origin along the x-axis. It's an essential step for comparing with limits along other paths.

Limit along the y-axis

Similarly to the x-axis, we evaluate limits by considering the path along the y-axis where \(x = 0\).
By substituting \(x = 0\) into the function, it simplifies to \(\frac{ ext{sin}^{2}(y)}{y^{2}}\). As \(y\) approaches zero, we use the identity \(\text{sin}(y) \approx y\) for small values of \(y\). This helps us to evaluate the limit, which becomes:

• \(\lim_{y \to 0} 1 = 1\)

Checking the limit along the y-axis gives a different value than along the x-axis, indicating how dependent the limit is on the path taken.

Squeeze Theorem

The Squeeze Theorem is a useful tool in calculus for finding limits by "squeezing" a function between two other functions that have the same limit. With multivariable functions and complex limits, this theorem helps to prove convergence when direct computation is difficult.
Although not directly used in our exercise, understanding this theorem bolsters intuition for limits. It involves three functions: if \(g(x, y) \leq f(x, y) \leq h(x, y)\) and \(\lim_{(x, y) \to (a, b)} g(x, y) = \lim_{(x, y) \to (a, b)} h(x, y) = L\), then \(\lim_{(x, y) \to (a, b)} f(x, y) = L\). This tool is significant when comparing complex paths or demonstrating that functions converge to a certain point from different directions.

Path dependence in limits

In multivariable functions, the overall limit can depend heavily on the path taken to approach a point. This path dependence was illustrated in our problem by comparing limits found along the x-axis and y-axis.
If the limit values differ, it conclusively shows that the overall limit does not exist at that point. In our given exercise, the limit along the x-axis was \(\frac{1}{2}\) and along the y-axis was 1, confirming no single limit exists as we reach \((0, 0)\).
Recognizing path dependence is crucial when working with multivariable limits. Not every path will necessarily yield a limit, making it essential to test along various lines or curves to determine the true nature of the function's behavior at specific points.