Problem 32 - Calculus for Biology and Medicine

Question

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0} \frac{\sqrt{2-x}-\sqrt{2}}{2 x} $$

Step-by-Step Solution

Answer

\( -\frac{1}{4\sqrt{2}} \)

Step 1: Analyzing the Expression

We are given the expression \( \lim_{x \rightarrow 0} \frac{\sqrt{2-x}-\sqrt{2}}{2x} \). This involves a division that approaches 0/0 as \( x \to 0 \), which suggests the limit may require special techniques to evaluate.

Step 2: Simplifying the Expression

Simplify the expression to better understand the behavior of the limit. Multiply the numerator and the denominator by the conjugate of the numerator: \( (\sqrt{2-x}+\sqrt{2}) \). This gives us:\[ \frac{(\sqrt{2-x}-\sqrt{2})(\sqrt{2-x}+\sqrt{2})}{2x(\sqrt{2-x}+\sqrt{2})} = \frac{(2-x) - 2}{2x(\sqrt{2-x}+\sqrt{2})} = \frac{-x}{2x(\sqrt{2-x}+\sqrt{2})} \].

Step 3: Cancelling Common Factors

Cancel the \( x \) factor from the numerator and the denominator:\[ \frac{-x}{2x(\sqrt{2-x}+\sqrt{2})} = \frac{-1}{2(\sqrt{2-x}+\sqrt{2})} \].

Step 4: Evaluating the Simplified Limit

Substitute \( x = 0 \) into the simplified expression since it's no longer in indeterminate form:\[ \lim_{x \to 0} \frac{-1}{2(\sqrt{2-x}+\sqrt{2})} = \frac{-1}{4\sqrt{2}} \].

Step 5: Verifying with Graphical Approach

To further verify, graph the function \( y = \frac{\sqrt{2-x}-\sqrt{2}}{2x} \) near \( x=0 \). Notice that as \( x \) approaches 0, the limit value of the graph stabilizes around \( -\frac{1}{4\sqrt{2}} \).

Key Concepts

Indeterminate Forms

When evaluating limits in calculus, you might encounter expressions that are not immediately straightforward to compute. One common case is the **indeterminate form**. This happens when the direct substitution of the value brings about expressions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), which do not offer a clear result. In the given exercise \( \lim_{x \rightarrow 0} \frac{\sqrt{2-x}-\sqrt{2}}{2x} \), you find yourself dealing with such an indeterminate form when substituting \( x = 0 \) leads to 0 in both the numerator and denominator. Dealing with indeterminate forms usually involves algebraic manipulation to simplify the expression, thus allowing you to find the limit by solving a simpler form that doesn't result in \( \frac{0}{0} \). This is why indeterminate forms are considered a signal that more work, often involving techniques like multiplication by conjugates or L'Hôpital's rule, is needed to resolve the limit.

Conjugate Method

The **conjugate method** is one of the algebraic techniques often used to tackle limits that initially present themselves as indeterminate forms. In our example, simplifying the expression \( \frac{\sqrt{2-x}-\sqrt{2}}{2x} \) requires a strategy. You multiply the numerator and the denominator by the conjugate of the numerator: \((\sqrt{2-x}+\sqrt{2})\). This clever trick transforms the numerator into a difference of squares, a commonly helpful algebraic identity: \((a-b)(a+b) = a^2-b^2\).

• This results in \((2-x)-2 = -x\) in the numerator, effectively canceling out the \( x \) factor that was causing the indeterminate quotient \( \frac{0}{0} \).
• Once simplified, you can cancel the \( x \) from both numerator and denominator, leading to a more solvable expression \( \frac{-1}{2(\sqrt{2-x}+\sqrt{2})} \).

By substituting \( x = 0 \), the expression no longer suffers from indeterminacy, allowing you to evaluate the limit smoothly and find \( \frac{-1}{4\sqrt{2}} \). The conjugate method is powerful in simplifying complex roots and evaluating limits.

Graphical Verification

After algebraically finding the limit, it’s beneficial to use graphical methods to confirm your solution. **Graphical verification** serves as a visual endorsement of the analytical work done. By plotting the function \( y = \frac{\sqrt{2-x}-\sqrt{2}}{2x} \) around \( x=0 \), you can observe how the function behaves as it approaches the limit point.

• In this scenario, graphing illustrates that as \( x \to 0 \), the function's value stabilizes at approximately \( -\frac{1}{4\sqrt{2}} \), reinforcing the result obtained through algebra.
• This approach not only supports the numerical solution but also strengthens understanding by providing a visual context.

Graphical verification is particularly helpful for those who benefit from a more intuitive feel of the function. It reassures that the mathematical derivations align with the expected real-world behavior of the function, thereby validating the analytical methods employed.