Problem 31 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is \( \frac{1}{2} \).

Step 1: Simplify the Expression

The limit to investigate is \( \lim _{x \rightarrow 0} \frac{1-\sqrt{1-x^{2}}}{x^{2}} \). This expression becomes indeterminate as \( x \to 0 \). To address this, consider multiplying the numerator and denominator by the conjugate of \( 1-\sqrt{1-x^2} \), which is \( 1+\sqrt{1-x^2} \).

Step 2: Multiply by the Conjugate

Multiply the given expression by \( \frac{1+\sqrt{1-x^2}}{1+\sqrt{1-x^2}} \) to eliminate the square root in the numerator:\[\frac{(1-\sqrt{1-x^2})(1+\sqrt{1-x^2})}{x^2(1+\sqrt{1-x^2})}\]The numerator simplifies to \( (1-x^2) - (1) = x^2 \), so the entire expression becomes:\[\frac{x^2}{x^2(1+\sqrt{1-x^2})}\]

Step 3: Simplify Further

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

Step 4: Evaluate the Limit

Substitute \( x = 0 \) into the simplified expression \( \frac{1}{1+\sqrt{1-x^2}} \):\[\lim _{x \rightarrow 0} \frac{1}{1+\sqrt{1-x^2}} = \frac{1}{1+\sqrt{1-0^2}} = \frac{1}{1+1} = \frac{1}{2}\]

Step 5: Confirm with a Table

For verification, create a table of values for the expression \( \frac{1-\sqrt{1-x^2}}{x^2} \) as \( x \) approaches 0, for very small positive and negative values of \( x \):- When \( x = 0.1 \), the function approximates \( 0.4997 \)- When \( x = -0.1 \), the function approximates \( 0.4997 \)As \( x \) approaches 0 from both sides, the values in the table approach \( 0.5 \).

Key Concepts

Indeterminate Forms

When dealing with limits in calculus, you may encounter expressions that result in forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These are known as indeterminate forms. They do not provide immediate information about what the limit might be. Indeterminate forms suggest that further exploration or manipulation is needed to evaluate the limit accurately. Examples of indeterminate forms include:

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)
• \( 0 \times \infty \)

To resolve an indeterminate form, techniques such as algebraic manipulation, L'Hôpital's rule, or rationalization are usually employed. Rationalizing the numerator or denominator, or both, can help transform the expression into a determinate form. In this specific exercise, the form \( \frac{0}{0} \) arises at \( x = 0 \), warranting the rationalization approach to simplify the calculation.

Rationalization Technique

Rationalization is a technique used to eliminate square roots from a fraction. This is particularly useful when evaluating limits that result in indeterminate forms. In the given problem, we applied the rationalization technique by multiplying the expression by the conjugate pair. The rationalization technique involves two main steps:

• Identify the square root in the expression.
• Multiply the numerator and denominator by the conjugate of the expression containing the square root.

For example, in this problem:\[\text{Original Expression: } \frac{1 - \sqrt{1 - x^2}}{x^2}\]We multiply by:\[\frac{1 + \sqrt{1 - x^2}}{1 + \sqrt{1 - x^2}}\]This step removes the square root from the numerator, simplifying the expression significantly and making it easier to determine the limit.

Conjugate Pairs

A crucial part of the rationalization technique is using conjugate pairs to simplify expressions involving square roots. A conjugate pair consists of two binomials that have the same terms but opposite operations between them. In mathematical terms, the conjugate of \( a - b \) is \( a + b \).When you multiply conjugate pairs together, it results in the difference of squares formula:\[(a - b)(a + b) = a^2 - b^2\]In the step-by-step solution, we used the conjugate of \( 1 - \sqrt{1-x^2} \), which is \( 1 + \sqrt{1-x^2} \). This operation simplifies the expression from:\[(1 - \sqrt{1-x^2})(1 + \sqrt{1-x^2}) = 1 - (1 - x^2) = x^2\]Applying this simplification removes the square root and leaves us with a manageable expression to evaluate.

Function Table Analysis

To confirm a limit evaluation, constructing a function table is a practical way to observe behavior as \( x \) approaches a specific value. It allows you to see numerical trends and verify analytical results. Here's how you can conduct a function table analysis:

• Choose values of \( x \) close to the target point on both sides (e.g., slightly negative and slightly positive values).
• Calculate the expression for each chosen value.
• Observe the output. The closer \( x \) values are to the limit point, the more precise the behavior you can study.If these results approach a particular value, you can be more confident in your analytical solutions.

In this problem, by choosing small values like \( 0.1 \) and \( -0.1 \), and calculating, we found the function returned values nearing \( 0.5 \), confirming that the limit as \( x \rightarrow 0 \) is indeed \( \frac{1}{2} \).