Question

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 0^{+}}\left(6-\frac{1}{x^{3}}\right) $$

Short answer

The one-sided limit of \(6-\frac{1}{x^{3}}\) as \(x\) approaches \(0\) from the positive side is \(-\infty\).

Step-by-step solution

Understand the Problem

In this step, we first understand the idea of a limit as \(x\) approaches \(0\) from the positive side. This means that we consider values of \(x\) that are very close to \(0\), but are slightly greater than \(0\). Mathematically, this is represented as \(\lim _{x \rightarrow 0^{+}}\), where the '+' superscript indicates that \(x\) is approaching \(0\) from the right or from positive values.

Substitute \(x\) in the Expression and Find the Limit

In the next step, we substitute \(x\) in the expression given and try to find the limit. The expression \(6-\frac{1}{x^{3}}\) becomes undefined at \(x = 0\), so we need to consider the behavior of this expression as \(x\) gets very close to \(0\) from the right. As \(x\) gets very close to \(0\) but is slightly greater than \(0\), \(\frac{1}{x^{3}}\) becomes very large because we are dividing 1 by a very small positive number. This makes the entire expression \(6-\frac{1}{x^{3}}\) very large in the negative sense. Therefore, the limit of this expression as \(x\) approaches \(0\) from the right is \(-\infty\).

State the Final Answer

Finally, we conclude that the one-sided limit of \(6-\frac{1}{x^{3}}\) as \(x\) approaches \(0\) from the positive side is \(-\infty\). This is because as \(x\) gets very close to \(0\) from the right, the expression \(6-\frac{1}{x^{3}}\) becomes very large in the negative sense. Therefore, the limit does not exist.

Key concepts

Limits in Calculus

Limits are fundamental concepts in calculus, helping us understand how functions behave as they approach a particular point, even when they don't actually reach that point. A limit looks at values coming from both sides of a number, or in some cases, just one side known as a one-sided limit.

In our exercise, we focus on a one-sided limit, which specifically examines how a function behaves for values approaching a point from one defined side - either the left or the right.

• Finding the Limit: We want to understand what happens to the function as the input values grow exceedingly close to a target number.
• One-Sided Limit: In one-sided limits like \(\lim _{x \rightarrow 0^{+}}(6-\frac{1}{x^{3}})\), we only consider values slightly greater than 0, signifying approach from the right.

Understanding limits involves intuition about the function's behavior near a point, which is crucial for mastering calculus.

Behavior of Functions Near Points

The behavior of functions as they approach certain points can reveal much about their characteristics, especially when studying limits. By studying the function's expression, we can predict how the function behaves when x moves in towards a specific value such as zero.

For instance, in our function \(6-\frac{1}{x^{3}}\), as x approaches 0 from the positive side, the behavior is largely determined by \(\frac{1}{x^{3}}\):

• Rapid Increase: As x becomes a very small number but still positive, \(\frac{1}{x^{3}}\) increases rapidly in value towards positive infinity.
• Dominant Negative Aspect: This means \(6-\frac{1}{x^{3}}\) produces very large negative outputs because we subtract a massive number from 6.

Being aware of this helps to predict that the function's value will drop endlessly below any fixed number, showcasing an approach toward negative infinity.

Undefined Expressions in Calculus

In calculus, an expression is termed 'undefined' if it leads to an undefined mathematical operation, such as division by zero. These undefined expressions often present themselves while evaluating limits.

When looking at our specific function, \(6-\frac{1}{x^{3}}\), as x reaches 0 from the positive side, there’s a division by a tiny number, catapulting the equation towards infinity or negative infinity, making it undefined.

• Behavioral Insight: Undefined outcomes help us signify the behavior of limits approaching certain critical points.
• Character of Infinity: Despite being undefined, we say the limit as x approaches 0 from the positive side is negative infinity, indicating unbounded behavior as opposed to a specific value.

Grasping these concepts ensures you adeptly handle similar undefined expressions while conducting limit evaluations in calculus.