Problem 19 - Essential Calculus

Question

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow-\infty} \frac{6 x^{2}}{x+3} $$

Step-by-Step Solution

Answer

The limit of the function as \(x\) approaches \(-\infty\) is \(-\infty\).

Step 1: Identification of the Highest Powers

Looking at the function \(\frac{6x^{2}}{x+3}\), we identify the highest power of x in the equation, which is \(x^{2}\) on the numerator.

Step 2: Division of Each Term by \(x^{2}\)

Divide every term in the equation by \(x^{2}\) to simplify the equation. This gives us \(\frac{6 \cdot 1}{\frac{x}{x^{2}}+\frac{3}{x^{2}}}\) which further simplifies to \(\frac{6}{\frac{1}{x}+\frac{3}{x^{2}}}\).

Step 3: Calculation of the Limit

Now to find the limit as \(x\) goes to \(-\infty\), we substitute \(x\) with \(-\infty\) in the simplified equation. The two terms in the denominator, \(\frac{1}{x}\) and \(\frac{3}{x^{2}}\), will approach zero as \(x\) goes to \(-\infty\). Thus the equation becomes \(\frac{6}{0}\).

Step 4: Dealing with Undefined Values

The situation of \(\frac{6}{0}\) is undefined, but in this case, we know that the ratio is becoming arbitrarily large because the denominator is approaching zero. Therefore, we can reason that the limit of the function as \(x\) goes to \(-\infty\) is \(-\infty\).

Key Concepts

Asymptotic Behavior

Understanding asymptotic behavior is key in calculus, especially when looking at how functions behave as variables approach infinity. Asymptotic behavior describes how a function behaves as it approaches a specific point or value, but never quite reaches it.
For example, when we analyze the function \( \frac{6x^2}{x+3} \) as \( x \) approaches \(-\infty\), we're interested in how the values of the function behave.
The function's numerator is growing rapidly while the denominator trends towards \(-x\), making the function's value increase in magnitude.

• When dealing with asymptotes, envision a line that the graph of a function tends to get closer and closer to, but does not need to actually meet.
• This is a foundational concept for determining the overall behavior and direction of a function as it approaches an extreme.

By looking at these behaviors, we can predict where the function is headed, even if it doesn't reach that exact point.

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a certain value. In the exercise, we see the notation \( \lim_{x \rightarrow -\infty} \), which means we're examining what happens to the function as \( x \) becomes very large negatively.
In such cases, terms like \( \frac{1}{x} \) and \( \frac{3}{x^2} \) shrink towards zero as \( x \) goes to \(-\infty\). This simplification is crucial for solving these limits.

• With infinite limits, the key is recognizing how specific terms behave when the input is very large or very small.
• For our function, as \( x \to \infty \), the numerator weighted by \( 6x^2 \) outpaces growth in the denominator, causing the function to extend towards an infinite result, in this case, \(-\infty\).

This understanding helps us predict the unbounded nature of the limit as it approaches its extreme value.

Rational Functions

Rational functions are fractions where both the numerator and denominator are polynomials. In the exercise, \( \frac{6x^2}{x+3} \) is our rational function. Analyzing these functions involves understanding the degrees of the polynomials involved.
The numerator has a degree of 2 because of \( 6x^2 \) and the denominator has a degree of 1 due to \( x \).

• If the numerator's degree is higher, as \( x \to \pm \infty \), the function tends towards infinity — negative or positive depends on the sign of \( x \).
• When the degree of the numerator equals the degree of the denominator, it tends towards a constant value. If smaller, it typically approaches zero.

Thus, understanding rational functions' behavior based on polynomial degrees is vital. This analysis allows for an accurate prediction of the function’s long-term behavior, leading to the determination of asymptotic behaviors and limits.