Question

Finding Functions Find functions \(f\) and \(g\) such that \(\lim _{x \rightarrow c} f(x)=\infty\) and \(\lim _{x \rightarrow c} g(x)=\infty,\) but \(\lim _{x \rightarrow c}[f(x)-g(x)] \neq 0\)

Short answer

The functions \(f(x) = 2x\) and \(g(x) = x\) fulfill the requirements of the exercise, given that we choose \(c\) different from zero. As \(x\) approaches \(c\), both these functions tend to infinity, while their difference \(f(x) - g(x) = x\) tends to \(c\) which is not zero.

Step-by-step solution

Choosing the functions

To fulfill these requirements, let's choose two simple functions, for example \(f(x) = 2x\) and \(g(x) = x\). As \(x\) approaches any value \(c\), both of these functions tend to infinity.

Checking the limit of the difference

To validate our choice, we now have to check that the limit of the difference between \(f(x)\) and \(g(x)\) as \(x\) approaches \(c\) is different from zero. The difference \(f(x) - g(x)\) turns out to be \(2x - x = x\). Now, when taking the limit of that difference as \(x\) tends to \(c\), we have \(\lim_{x \rightarrow c} [f(x) - g(x)] = \lim_{x \rightarrow c} x = c\). This limit is not zero, unless we specifically choose \(c\) to be zero. Therefore, if we choose \(c\) different from zero, the difference between the functions \(f(x)\) and \(g(x)\) doesn't equal to zero in the limit.

Key concepts

Infinite Limits

In calculus, the concept of infinite limits refers to the behavior of a function as its value grows without bound, either positively or negatively, when the independent variable approaches a certain point. For instance, as you examine the function \( f(x) = 2x \), as \( x \) gets larger and larger, the value of the function also increases without end. Similarly, \( g(x) = x \) behaves this way as well.

Think of it like a rocket soaring into space; as time goes on, it just keeps going higher, representing an infinite limit. When \( x \) approaches a constant value \( c \), and the function heads to infinity, you would write this as \( \text{{lim}}_{x \rightarrow c} f(x)=\text{{\textbackslash}}infty \). The concept is a fundamental part of understanding how functions behave near specific points and is crucial for analyzing the convergence or divergence of series, evaluating integrals, and solving differential equations.

It's important to note that even when individual functions approach infinity, the difference between functions can still yield diverse and meaningful results, an aspect we'll delve into in the next section.

Difference of Functions

When dealing with the difference of functions, we are essentially examining what happens when one function is subtracted from another. Crucially, when both functions independently approach infinity, the difference does not automatically result in a zero or infinity; it depends on the relationship between the growth of the two functions.

Here's an illustration using the functions from the exercise: \( f(x) = 2x \) and \( g(x) = x \). As \( x \) approaches any value \( c \), individually, both \( f(x) \) and \( g(x) \) trend toward infinity. However, when we calculate the limit of the difference \( f(x) - g(x) \), we get a new function \( h(x) = x \), which is not tending towards infinity, but rather to the specific value \( c \).

This elucidates that the behavior of a function created by the difference of two other functions can have unique properties and is not always intuitive. Understanding these relationships is key to mastering concepts in calculus, such as determining the convergence of series or the behavior of functions at specific points.

Calculus Concepts

Calculus is an area of mathematics that studies how things change; it provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models. Two big pieces of calculus are differential calculus, which concerns the concept of a derivative, depicting the rate of change of a quantity, and integral calculus, which focuses on the concept of an integral, representing the accumulation of quantities.

Functions, limits, and differences like those in the given exercise are foundational concepts within calculus. The ability to examine the limit of a function as it approaches a specific value gives rise to derivatives. In examining the difference between two functions, as we did with the functions \( f(x) = 2x \) and \( g(x) = x \), we utilize these basic tools to move towards more complex applications, such as finding areas under curves, the total accumulation of a function, or even solving complex real-world problems that involve rates of change.

It's crucial for students to build a strong foundational understanding of these concepts, as they serve as building blocks for more complex and advanced topics in calculus and are widely applicable in various fields of science and engineering.