Problem 53 - Calculus of a Single Variable

Question

Writing In Exercises 53 and \(54,\) describe in your own words what the statement means. $$ \lim _{x \rightarrow \infty} f(x)=4 $$

Step-by-Step Solution

Answer

The statement means that as x grows bigger and bigger without any upper limit, the value of the function f(x) approaches 4.

Step 1 Understanding Limit

A limit in mathematics usually describes the behaviour of a function when its variable approaches a certain value. In the given exercise, the limit of function f(x) is given as x approaches infinity.

Step 2 Understanding Infinity

Infinity, denoted by the symbol '∞', is not a real number but a concept meaning 'endless' or 'limitless' in mathematics. When we say that x approaches infinity, it means that x is increasing without bound.

Step 3 Interpret the Statement

Given the understanding of a limit and the concept of infinity, we can interpret the given statement. It means when the input value x in the function f(x) grows larger and larger without any upper bound, the function value or the output of that function approaches and stabilizes at the value 4.

Key Concepts

Limit of a Function

The concept of limit of a function is a cornerstone in calculus and involves predicting the value that a function approaches as the input (or 'x value') gets closer and closer to a specific point. It's important to understand that a limit isn't necessarily the value the function actually reaches, but it is the value the function strives to achieve.

To put it simply, imagine you're walking towards a tree. The tree represents the limit—the value that you're trying to reach. As you take steps forward, you're the equivalent of 'x' getting closer to the 'point'. Even if you decide to stop before you reach the tree, you know the direction you were heading in was towards the tree. Similarly, in calculus, we aren't always concerned with where 'x' actually is but with where it is headed as it approaches a certain value.

Consider the function f(x) = (2x + 3)/(x + 1). As 'x' gets larger and larger, the terms without 'x' become less significant, and we can say that the function approaches the limit of 2 since (2x + 3)/(x + 1) approaches 2 when x goes towards infinity. This doesn't mean the function will ever be equal to 2 for some value of x; rather, it gets indefinitely close to 2 as 'x' becomes larger.

Approaching Infinity

The phrase approaching infinity might seem abstract at first glance, but it's a pivotal idea in understanding the behavior of functions in calculus. Infinity, symbolized by \(\infty\), represents an unbounded limit, a value larger than any finite number, or a notion of endlessness.

When we say an expression like \(x \rightarrow \infty\), we imply that 'x' grows larger without any bounds—imagine a number line stretching forever and 'x' traveling along it, heading towards the right without end. It never actually arrives at infinity because infinity is not a place or a number; it's the concept of 'without end'.

Take the function \(f(x) = 1/x\). As 'x' increases (\(x \rightarrow \infty\)), the fraction \(1/x\) gets smaller and smaller, approaching zero. It never quite reaches zero; no matter how large 'x' gets, \(1/x\) is always just a little bit greater than zero. This shows how a function can approach a finite limit even as the variable approaches infinity.

Understanding Mathematical Limits

To truly grasp mathematical limits, it's important to recognize that limits allow us to discuss and predict the behavior of functions at points that might not be directly assessable. This could be due to the function being undefined at that point, or because the point involves reaching towards an infinite boundary.

In the given exercise \(\lim _{x \rightarrow \infty} f(x)=4\), we see the statement means as 'x' becomes larger and larger—seemingly without any restriction—the function \(f(x)\) settles towards the value 4. This doesn't mean it touches or is equal to 4 but that its values get infinitely close to 4.

An analogy might be the temperature outside approaching room temperature as evening falls. Even though the precise moment the outside temperature is exactly room temperature might be fleeting or theoretically imperceptible, we can understand that it moves towards and hovers around that temperature. Much like this, mathematical limits describe the 'approaching' rather than the 'reaching', painting a picture of a function's tendencies near certain points or as variables embark on the march towards infinity.