Problem 76 - Calculus Volume 3

Question

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{\sqrt{x^{2}+y^{2}+1}-1}$$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Simplify the Expression

To evaluate \( \lim_{(x, y) \to (0, 0)} \frac{x^2 + y^2}{\sqrt{x^2 + y^2 + 1} - 1} \), we notice the expression involves an indeterminate form at \((0,0)\). A common technique for such expressions involves rationalizing the denominator. Multiply the numerator and the denominator by \(\sqrt{x^2+y^2+1} + 1\) to rationalize:\[ \frac{(x^2 + y^2)(\sqrt{x^2 + y^2 + 1} + 1)}{(\sqrt{x^2 + y^2 + 1} - 1)(\sqrt{x^2 + y^2 + 1} + 1)} = \frac{(x^2 + y^2)(\sqrt{x^2 + y^2 + 1} + 1)}{x^2 + y^2}\]This simplification uses the identity \((a-b)(a+b) = a^2 - b^2\).

Step 2: Cancel Out Common Terms

After rationalizing, the expression simplifies to:\[ (\sqrt{x^2 + y^2 + 1} + 1)\]The \(x^2 + y^2\) terms in the numerator and denominator cancel each other out as long as \((x, y) eq (0,0)\).

Step 3: Evaluate the Limit at (0,0)

Now, evaluate the limit as \((x, y) \to (0, 0)\) of the simplified expression:\[ \lim_{(x, y) \to (0, 0)} (\sqrt{x^2 + y^2 + 1} + 1) = \sqrt{0^2 + 0^2 + 1} + 1 = \sqrt{1} + 1 = 2\]Thus, the original limit evaluates to 2.

Key Concepts

Limits in Two Variables

In multivariable calculus, understanding limits in two variables involves approaching a point in the coordinate plane from various directions. It's similar to single-variable limits, but with the added complexity of a plane. Rather than just moving along a line, you might approach a specific point from any angle.

Consider the function given in the problem: \[\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{\sqrt{x^{2}+y^{2}+1}-1}\]Here, we look at how both variables `x` and `y` simultaneously approach zero. To confirm if a limit exists, we need the same result irrespective of the path taken towards `(0,0)`. This path independence is crucial for the limit to exist. The expression represents an approach where considering specific paths can help, like suggesting polar coordinates to simplify and visualize the convergence.

Understanding the concept also requires discerning that some limits might not exist due to different results from different paths, indicating a specific point's undefined nature.

Indeterminate Forms

Indeterminate forms often occur in calculus when substituting a point into a limit expression does not produce a clear answer. Common types include \(\frac{0}{0}\) and \(\infty - \infty\).

In the given exercise, substituting `(0,0)` into the function produces a \(\frac{0}{0}\) form. This signals the need for further manipulation to resolve the ambiguity.

• First, check the form by direct substitution.
• Next, if indeterminate, look for algebraic strategies to simplify.

These steps are essential in determining paths towards finding real limit values. Given situations like this often require using different strategies such as algebraic simplifications, factorizations, or rationalizations, as applied in our original problem by turning the denominator into a more manageable form.

Rationalizing Techniques

Rationalizing is a powerful method used to simplify expressions, particularly useful when dealing with limits involving square roots or other roots. The technique involves multiplying by a form of 1 to clear a root from the denominator.

For the problem\[\frac{x^{2}+y^{2}}{\sqrt{x^{2}+y^{2}+1}-1},\]we encounter the square root in the denominator, making simplification necessary to resolve the limit at \( (0,0) \).

• Multiply the numerator and denominator by the conjugate of the denominator: \(\sqrt{x^2+y^2+1} + 1\).
• This manipulation clears the root from the denominator resulting in a simplified expression.

This step utilizes the identity \((a-b)(a+b) = a^2 - b^2\) to transform the expression into a more approachable form. Simplifying these expressions aids in evaluating the limit accurately and effectively, leading to the easier calculation of the actual limit value as demonstrated, which leads to the correct evaluation of \(2\) at the limit's point of focus.