Problem 44 - Calculus for Biology and Medicine

Question

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 1} \frac{1}{\sqrt{3-2 x^{2}}} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Analyze the Function

The function given is \( f(x) = \frac{1}{\sqrt{3-2x^2}} \). We need to find \( \lim_{x \to 1} f(x) \). First, check if the function is defined at \( x = 1 \). Substitute \( x = 1 \) into the expression inside the square root: \( 3-2(1)^2 = 3-2 = 1 \). Since the square root of 1 is defined, \( f(x) \) is defined at \( x=1 \).

Step 2: Substitute the Limit Point

Now substitute \( x = 1 \) into the function to find the limit: \[ \lim_{x \to 1} \frac{1}{\sqrt{3-2x^2}} = \frac{1}{\sqrt{3-2(1)^2}} = \frac{1}{\sqrt{1}}. \]

Step 3: Simplify the Expression

Simplify the expression found in Step 2. \( \sqrt{1} = 1 \), so the expression becomes \( \frac{1}{1} = 1 \).

Step 4: Finalize the Limit

Conclude that the limit of \( f(x) \) as \( x \) approaches 1 is 1. Therefore, \( \lim_{x \to 1} \frac{1}{\sqrt{3-2x^2}} = 1 \).

Key Concepts

Limit Evaluation

Evaluating a limit helps us understand the behavior of a function as it approaches a particular point. In this exercise, we calculate the limit \( \lim_{x \to 1} \frac{1}{\sqrt{3-2x^2}} \). Often, the first step in limit evaluation is to analyze if the function is defined at the limit point.
For our case, we substitute \( x = 1 \) into the expression under the square root: \( 3-2(1)^2 = 1 \). The square root is defined here (since \( \sqrt{1} = 1 \)), allowing us to proceed.
Once we determine the function is defined, we substitute \( x = 1 \) into the entire function:

• Evaluate: \( \frac{1}{\sqrt{3-2(1)^2}} = \frac{1}{1} \)


• Simplify: \( \frac{1}{1} = 1 \)

This process confirms that the limit exists and equals 1, showcasing the simplicity and beauty behind evaluating limits.

Square Roots

The square root function appears in many mathematical contexts and is crucial to understanding this problem. When dealing with square roots, especially within limits, it's essential to ensure the expression under the root is non-negative, as square roots of negative numbers are not defined in the real number system.
In our exercise, the expression within the square root is \( 3-2x^2 \). At \( x=1 \), this simplifies to \( 1 \), confirming that the square root is valid since it's a non-negative number. If at any point, this expression becomes less than zero, the function would be undefined, which would significantly alter our approach to finding the limit.
Always remember that understanding how a function behaves is vital. In cases where the expression under the root approaches zero, the limit might involve more nuanced mathematical techniques. However, for our function, the square root simplifies directly to an integer at \( x=1 \), making the process straightforward.

Function Analysis

Analyzing a function involves understanding its components and behavior as various inputs approach certain values. Here, we look at the rational function \( f(x) = \frac{1}{\sqrt{3-2x^2}} \). To perform a comprehensive function analysis, consider:

• The domain: Ensure all values under the square root are non-negative (i.e., solve \( 3-2x^2 \geq 0 \)).
• Continuity: Check if the function behaves smoothly at and around the point of interest, \( x=1 \).


• Asymptotic behavior: Observe if the function approaches a finite value (e.g., \( \frac{1}{1} = 1 \) near \( x=1 \)).

Through function analysis, we verify that our approach for evaluating the limit is appropriate. We identify the behavior and continuity at critical points. Understanding these principles assists in predicting behaviors and solving such problems accurately in calculus.