Problem 11 - Calculus for Biology and Medicine

Question

Evaluate the limits in problems. $$ \lim _{x \rightarrow-\infty} \frac{2+x^{2}}{1-x^{2}} $$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Understanding the Problem

We need to evaluate the limit of the function \( f(x) = \frac{2+x^2}{1-x^2} \) as \( x \to -\infty \). This involves finding the behavior of the fraction as \( x \to -\infty \).

Step 2: Rewriting the Expression

To simplify the expression, divide every term by \( x^2 \) to make the dominant terms evident as \( x \to -\infty \). Thus, the expression becomes \( \frac{\frac{2}{x^2} + 1}{\frac{1}{x^2} - 1} \).

Step 3: Simplifying for Large \( x \)

As \( x \to -\infty \), the terms \( \frac{2}{x^2} \) and \( \frac{1}{x^2} \) both tend towards zero because they are divided by \( x^2 \), which becomes very large. This simplifies the expression to \( \frac{0 + 1}{0 - 1} = \frac{1}{-1} \).

Step 4: Final Calculation

The resulting simplified expression is \( -1 \). Therefore, the limit of \( \frac{2+x^2}{1-x^2} \) as \( x \to -\infty \) is \( -1 \).

Key Concepts

Rational Functions

Rational functions are expressions that feature the ratio of two polynomials. These types of functions are expressed as \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. In our exercise, \( P(x) = 2 + x^2 \) and \( Q(x) = 1 - x^2 \).

• Numerator \( P(x) \) is \( 2 + x^2 \), a quadratic polynomial.
• Denominator \( Q(x) \) is \( 1 - x^2 \), also a quadratic polynomial.

Understanding the structure of rational functions is crucial for analyzing their behavior, especially around critical points where the polynomial in the denominator could become zero. This might lead to undefined or asymptotic behavior.

Asymptotic Behavior

Asymptotic behavior examines how functions behave as the variable approaches a specific value or approaches infinity. In the case of a rational function like \( f(x) = \frac{2+x^2}{1-x^2} \), we analyze the behavior of the function as \( x \to -\infty \).

• When \( x \) is very large (positively or negatively), the terms with the highest exponent in \( P(x) \) and \( Q(x) \) dominate.
• The behavior at \( x \to -\infty \) is determined by how the dominant terms interact, often simplifying to a more straightforward relationship.

For the given function, both numerator and denominator have dominant terms of \( x^2 \). This means they will significantly influence the limit as \( x \to -\infty \), resulting in an asymptote at the result of the limit. This understanding is essential for drawing accurate graphs and predicting function behavior.

Dominant Terms

Dominant terms in a polynomial are those with the highest power because they have the greatest influence as the variable grows larger. In the function \( f(x) = \frac{2+x^2}{1-x^2} \), the dominant terms are the \( x^2 \) terms.

• When \( x \to -\infty \), terms such as \( \frac{2}{x^2} \) and \( \frac{1}{x^2} \) become negligible since they tend to zero.
• This simplification reduces the fraction to approximately \( \frac{1}{-1} \), since other terms lose significance.

Understanding which terms dominate allows us to see the limit behavior quickly and predict outcomes for very large or small values of \( x \). It simplifies complex expressions and sheds light on their fundamental characteristics, which is particularly useful for limits and asymptotes.