Problem 3 - Calculus for Biology and Medicine

Question

Evaluate the limits in problems. $$ \lim _{x \rightarrow-\infty} \frac{x^{3}+3}{x-2} $$

Step-by-Step Solution

Answer

The limit approaches \(-\infty\).

Step 1: Understanding the Limit Expression

We need to evaluate the limit \( \lim_{x \rightarrow -\infty} \frac{x^3 + 3}{x - 2} \). This means we have to analyze the behavior of the function \( \frac{x^3 + 3}{x - 2} \) as \( x \) approaches negative infinity.

Step 2: Factor Out the Dominant Term in the Numerator and Denominator

The dominant term in both the numerator and denominator as \( x \to -\infty \) are the \( x^3 \) in the numerator and \( x \) in the denominator. We can factor \( x \) from the denominator and \( x^3 \) from the numerator to simplify: \[ \frac{x^3(1 + \frac{3}{x^3})}{x(1 - \frac{2}{x})} \].

Step 3: Simplify the Expression

Simplify the expression from the previous step: \( \frac{x^2 (1 + \frac{3}{x^3})}{(1 - \frac{2}{x})} \). Notice that as \( x \rightarrow -\infty \), \( \frac{3}{x^3} \) and \( \frac{2}{x} \) both approach zero, so the expression simplifies to \( \frac{x^2}{1} \).

Step 4: Evaluate the Dominant Behavior as \( x \rightarrow -\infty \)

Since \( x^2 \) grows larger and positive as \( x \) approaches \(-\infty\), the expression approaches infinity. Therefore, the limit \( \lim_{x \rightarrow -\infty} \frac{x^3 + 3}{x - 2} \to -\infty \) because \( x^2 \) is positive but \( x \) is negative.

Key Concepts

Dominant Term

When evaluating limits, especially as functions approach infinity or negative infinity, it is crucial to identify and focus on the dominant term. The dominant term is the term in the expression that grows the fastest and thus has the greatest influence on the behavior of the function in the limit.

• For example, in the expression \( x^3 + 3 \), the term \( x^3 \) is dominant because it increases much more rapidly than the constant \( 3 \) as \( x \) grows large.
• Similarly, in the denominator \( x - 2 \), the term \( x \) is the dominant term since it will overshadow the influence of \( -2 \) as \( x \) becomes very large or very small.

To simplify such expressions in limits involving infinity, we focus on these dominant terms to approximate the function's behavior at extreme values.

Infinity in Limits

Understanding limits as \( x \) approaches infinity, or negative infinity, involves analyzing the end behavior of a function. As \( x \) heads towards negative infinity for our current function, we deal with very large negative values.

• The notion of infinity in limits helps us describe how functions behave when \( x \) is not confined to typical values, but extends to bounds that are conceptually endless.
• This concept might bring the function either to very large positive numbers, infinite growth, or shrink to very large negative numbers, infinite decay.

In this exercise, since the dominant term \( x^3 \) in the numerator grows very large, it will have the primary impact on the fraction's value as \( x \) dives into negative infinity.

Behavior of Functions at Infinity

The behavior of a function as \( x \) approaches infinity, or in this case negative infinity, can be understood by studying its dominant terms, as they dictate the overall trend.

• In our example, after simplification, the function reduces to an expression dominated by \( x^2 \) due to factoring out the dominant \( x^3 \) from the original function.
• This reveals that as \( x \) moves to negative infinity, \( x^2 \) remains positive because squaring any negative number results in a positive value.
• Thus, the negative infinity behavior of the original expression translates into minus infinity because the growing positive number is in the numerator while \( x \) is originally negative.

Understanding this concept helps predict whether functions will shoot up to infinity or down to negative infinity when \( x \) approaches very large positive or negative values.