Question

Define (a) \(\lim _{x \rightarrow x_{0}} f(x)=\infty\) (b) \(\lim _{x \rightarrow x_{0}} f(x)=-\infty\)

Short answer

Question: Define the concepts of limit of a function as it approaches positive infinity and negative infinity at a particular point \(x_{0}\). Answer: The limit of a function \(f(x)\) as \(x\) approaches a particular point \(x_{0}\) equals positive infinity if for any large positive number \(M\), there exists a \(\delta > 0\) such that if \(0 < |x - x_{0}| < \delta\), then \(f(x) > M\) (expressed as \(\lim_{x \rightarrow x_{0}} f(x) = \infty\)). The limit equals negative infinity if for any large negative number \(M\), there exists a \(\delta > 0\) such that if \(0 < |x - x_{0}| < \delta\), then \(f(x) < M\) (expressed as \(\lim_{x \rightarrow x_{0}} f(x) = -\infty\)).

Step-by-step solution

Definition of Limit as the function approaches Positive Infinity

We say that the limit of a function \(f(x)\) as \(x\) approaches a particular point \(x_{0}\) equals positive infinity, if for any arbitrary large positive number \(M\), there exists a \(\delta > 0\) such that if \(0 < |x - x_{0}| < \delta\), then \(f(x) > M\). This is expressed mathematically as: $$\lim_{x \rightarrow x_{0}} f(x) = \infty$$

Definition of Limit as the function approaches Negative Infinity

We say that the limit of a function \(f(x)\) as \(x\) approaches a particular point \(x_{0}\) equals negative infinity, if for any arbitrary large negative number \(M\), there exists a \(\delta > 0\) such that if \(0 < |x - x_{0}| < \delta\), then \(f(x) < M\). This is expressed mathematically as: $$\lim_{x \rightarrow x_{0}} f(x) = -\infty$$

Key concepts

Limit of a Function

In mathematics, particularly in real analysis, the limit of a function is a fundamental concept that describes the behavior of that function as its argument approaches a certain point. It helps us understand what value a function approaches, even if it doesn't actually reach that value exactly.
For example, consider a function which grows larger and larger as it gets closer to the point from the left or right, without ever stopping. This idea can be tricky since functions do not always behave in a simple or predictable way as they approach a point. Therefore, we use limits to rigorously define and predict such behavior.
A limit is defined for a function at a certain point, and it can be a real number, positive infinity, or negative infinity. When teaching this concept, it's beneficial to imagine the function's graph and think how it behaves near the point of interest. Exercises that encourage plotting the function can be helpful for visual learners and provide additional clarification.

Positive Infinity

When dealing with the concept of positive infinity, denoted by the symbol \(+\infty\), we are talking about values that grow larger than any real number we can assign. It's not a number per se but a way to describe unbounded growth.
In terms of limits, if we say that the limit of a function is positive infinity as the variable approaches a certain point, it means that the function's value will increase without bound as it gets closer to that point. The formal definition used in textbooks involves an arbitrary large positive number, which represents the idea that no matter how large of a number you choose, the function will eventually exceed that value as it approaches the point in question. This behavior is characteristic of functions with vertical asymptotes or certain types of singularities.

Visualizing Positive Infinity

Imagine the graph of a function that keeps going up as it nears the point on the x-axis. The line never stops or levels off—it just keeps climbing towards the sky. That's essentially what positive infinity is capturing: limitless increase.

Negative Infinity

Conversely, the term negative infinity, which is symbolized as \(-\infty\), refers to values that keep decreasing beyond every extent of lessening we can conceive. It's the opposite of positive infinity and represents unbounded decrease.
When we define a function's limit as negative infinity, as the variable approaches a certain point, it signifies that the function will descend deeper and deeper as it nears that point. Like positive infinity, this concept is also translated into a formal definition using an arbitrary large negative number in the textbooks, reinforcing the concept that the function’s value will continue to fall, no matter how low you set the bar, as it gets closer to a predefined point.

Visualizing Negative Infinity

Think of the graph of a function that drops lower and lower as it approaches a point on the x-axis. Just like a bottomless pit, the line keeps sinking without ever hitting a floor. That's what negative infinity symbolizes: boundless decrease.

Real Analysis

The field of real analysis is a significant branch of mathematical analysis dealing with the behavior and properties of real numbers, sequences, series, and real-valued functions. It lays down the rigorous foundations for calculus, which most students are familiar with from high school.
Real analysis deals in detail with concepts like limits, continuity, differentiation, and integration. It defines them with great precision, which allows mathematicians to handle functions and sequences that behave in complex ways. The concepts of positive and negative infinity are deeply rooted in real analysis and are critical for understanding the behavior of functions at their extremes.
The subject might seem abstract at first, but real-world applications are abundant, from physics to engineering and beyond. Exercises in real analysis can often be greatly improved with graphical examples, helping to bridge the gap between abstract theory and intuitive understanding.