Question

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

Short answer

Question: Define the two different types of limit problems involving infinity. Answer: The two different types of limit problems involving infinity are: 1. \(\lim_{x \rightarrow \infty} f(x) = \infty\): This represents that as \(x\) becomes larger and larger (approaches infinity), the value of the function f(x) also becomes larger and larger (approaches infinity). It signifies that the function grows without bound as x increases. 2. \(\lim_{x \rightarrow -\infty} f(x) = -\infty\): This represents that as \(x\) becomes smaller and smaller (approaches negative infinity), the value of the function f(x) also becomes smaller and smaller (approaches negative infinity). It signifies that the function decreases without bound as x decreases.

Step-by-step solution

Understanding the concept of limits

A limit represents the value a function approaches as the input approaches a certain value. In this case, our input is x, and we want to know how the function f(x) behaves as x approaches infinity and negative infinity, respectively.

Define the limit as x approaches infinity

We have to define what is meant by \(\lim_{x \rightarrow \infty} f(x) = \infty\). This means that as \(x\) becomes larger and larger (approaches infinity), the value of the function f(x) also becomes larger and larger (approaches infinity). More formally, for any real number M, there exists a real number N such that if \(x > N\), then \(f(x) > M\). It signifies the function grows without bound as x increases.

Define the limit as x approaches negative infinity

Now, we have to define what is meant by \(\lim_{x \rightarrow -\infty} f(x) = -\infty\). This means that as \(x\) becomes smaller and smaller (approaches negative infinity), the value of the function f(x) also becomes smaller and smaller (approaches negative infinity). More formally, for any real number M, there exists a real number N such that if \(x < N\), then \(f(x) < M\). It signifies the function decreases without bound as x decreases.

Key concepts

Limit of a Function

To fully appreciate the intricacies of calculus and real analysis, it's essential to grasp the concept of the limit of a function. Imagine you're on a path that winds towards a mountain. As you move closer, your perspective of the peak becomes clearer — this is akin to a function whose output gets closer to a specific value as the input nears a certain point. The limit captures this nearing behavior in a rigorous mathematical sense.

In simpler terms, the limit tells us the value that a function is approaching as its input gets either very large or very close to some number. For instance, if you're examining the temperature throughout a day, the limit of the temperature function as time approaches noon gives you an idea of the temperature around that time, even if there's no exact temperature reading at noon itself.

• When the function continually increases or decreases without any upper or lower bound, the limit expresses this unbounded behavior, but instead of approaching a finite number, it reaches towards infinity or negative infinity.
• Understanding limits is not just about the end behavior but also about grasping how functions behave around points that might not be explicitly defined, such as holes or jumps in the function's graph.

Infinity in Limits

In the realm of limits, the concept of infinity is not just an idea from science fiction or philosophy, but a fundamental aspect we deal with mathematically. When we say that the limit of a function is infinity, it reflects an unbounded increase as the input grows without limit. Think of it like a rocket that continues to zoom through space, surpassing planets and stars, with no intention of stopping.

In mathematical terms, this is expressed as \(\lim_{x \rightarrow \infty} f(x) = \infty\)

• This implies that no matter how high you set a bar (any real number M), the function can provide you with a value that exceeds that bar.
• It's important to note that infinity is not a number we can reach; it's a concept that tells us the function doesn't settle at any finite value.

Behavior of Functions at Infinity

Exploring the behavior of functions at infinity is like observing the trajectory of a spaceship as it disappears into the cosmos; the path it takes towards infinity tells us a great deal about the overall nature of the function. Mathematically, this understanding helps to predict trends and behavior of the function when the input values become extremely large (positive or negative).

Functions can show different behaviors as they approach infinity:

• They may grow indefinitely, like savings in an account with continuous compound interest.
• Functions might decrease without bound, akin to the temperature as one delves into the depths of space.
• Some functions will approach a specific value, never quite reaching it but getting infinitesimally close, like a spacecraft stabilizing at a near-zero speed after running out of fuel.
• Others will oscillate endlessly, similar to a pulsar that emits beams of electromagnetic radiation with a regular rhythm.

This behavior gives rise to the concepts of horizontal asymptotes and end behavior models, which are critical in the analysis of functions and their limits.

Formal Definition of Limits

Diving deeper into the subject, the formal definition of limits is the backbone of understanding how functions behave as inputs approach certain values or infinity. Also known as the 'Epsilon-Delta' definition, it provides a mathematical framework for defining limits with precision and is crucial for proofs in calculus and analysis.

Here's a simplified depiction of the formal definition for approaching a finite value: For every tiny positive number (epsilon), there's a corresponding small range (delta) around the input value such that when the input is within this delta range, the output of the function will be within the epsilon range of the limit value. This definition nails down the concept of a function 'getting close' to a specific value.

• When dealing with infinity, this formalism adapts to accommodate the boundless nature. In the case of our exercise, it means that for any large number (M), there's an input threshold (N) beyond which all function values exceed M (when dealing with positive infinity) or fall below M (for negative infinity).
• This rigorous approach allows mathematicians to work with concepts like infinity in a concrete way, ensuring that the infinite behaviors are well-defined and analyzable within the realm of real numbers and functions. It is essential for higher-level understanding and working with theorems in calculus.