Problem 20 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(19-38,\) find the limit. $$ \lim _{x \rightarrow-\infty}\left(\frac{5}{x}-\frac{x}{3}\right) $$

Step-by-Step Solution

Answer

As \(x\) approaches \(-\infty\), the limit of the function is \(+\infty\).

Step 1: Rewrite the Function

Rewrite the function as a single fraction to simplify. As such, it is: \(-\frac{x}{3} + \frac{5}{x}\).

Step 2: Evaluate the Limits

Evaluate both terms separately. As \(x\) approaches \(-\infty\), the first term \(-\frac{x}{3}\) approaches positive infinity since a big negative number divided by 3 is still a large negative number (and the negative sign in front makes it positive). The second term, \frac{5}{x}, approaches 0, because any number divided by an increasingly large number tends to 0.

Step 3: Combine the Results

Combine the results of the two limits. As \(x\) approaches -\infty, the overall limit is \(+\infty + 0 =+\infty\).

Key Concepts

Limits at Infinity

When dealing with limits at infinity in calculus, we are essentially exploring the behavior of a function as the independent variable, typically denoted as 'x', either goes to positive or negative infinity. This gives us insight into how the function behaves at extreme ends of its domain.

As an example, consider the function \( \lim_{x \rightarrow -\infty} \left( \frac{5}{x} - \frac{x}{3} \right) \). To understand this limit, we need to consider the rate at which each part of the function grows or decays. The expression \( \frac{5}{x} \) suggests that as 'x' becomes very large in the negative direction, the value of the fraction will get closer to zero, symbolically, this part would tend to \(0\).

On the other hand, for the expression \( -\frac{x}{3} \), as 'x' goes to negative infinity, the value grows positively infinite because a large negative value divided by a smaller negative number yields a big positive number. Limit evaluation, as 'x' tends to \( -\infty \), involves monitoring these rates of change and understanding that, in this case, the dominant term \( -\frac{x}{3} \) dictates that the limit tends to positive infinity.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus that involves finding the value that a function approaches as the input approaches a particular point. When 'x' is moving toward a finite number, we seek the function's value at that number. When 'x' is approaching infinity, the evaluation differs significantly as we're more concerned with the function's trend or end behavior.

Evaluating \( \lim_{x \rightarrow c} f(x) \) where 'c' could be any real number, or even \( \pm \infty \) involves a variety of techniques. One can factorize, rationalize, or even rewrite functions to simplify them. For instance, the given function \( \( -\frac{x}{3} + \frac{5}{x} \) \) was rewritten as a single term to easily analyze both \( -\frac{x}{3} \) and \( \frac{5}{x} \) separately. This rewrite makes it clear that as 'x' becomes very large (in the positive or negative sense), the term with 'x' in the denominator tends to \(0\), while the linear term determines the limit's trend.

Negative Infinity in Limits

Limit calculations that involve negative infinity occur when 'x' is heading towards negative infinity. The convention of negative infinity affects the limit's behavior just as positive infinity does, but often in an opposite manner.

In the case of the function \( \( -\frac{x}{3} + \frac{5}{x} \) \) as 'x' approaches \( -\infty \), we expect the terms that involve 'x' in the numerator to increase without bound in the negative direction. However, because of the negative sign in front of the term \( -\frac{x}{3} \) in our specific example, the 'negative' negative becomes a 'positive.' As a result, the limit as 'x' approaches negative infinity is not just 'infinity,' but specifically positive infinity.

This subtle yet essential distinction is vital in understanding how we evaluate the limits that involve negative infinity. Although we cannot assign a numerical value to infinity, the direction of the trend—to positive or negative infinity—provides crucial information about the function's end behavior.