Problem 6 - Precalculus: Mathematics for Calculus

Question

Find the limit. $$\lim _{x \rightarrow \infty} \frac{3}{x^{4}}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Limit Expression

The task is to find the limit of the function \( \frac{3}{x^4} \) as \( x \) approaches infinity. This means we need to evaluate what happens to \( \frac{3}{x^4} \) when \( x \) becomes very large.

Step 2: Analyze the Function Behavior

Consider the function \( \frac{3}{x^4} \). As \( x \) becomes larger, the denominator \( x^4 \) becomes significantly larger, causing the overall value of the fraction to decrease.

Step 3: Apply Limit Concepts

Since the denominator \( x^4 \) grows without bound as \( x \to \infty \), the fraction \( \frac{3}{x^4} \) approaches zero. Essentially, \( \frac{3}{x^4} \to 0 \) as \( x \to \infty \).

Step 4: State the Final Result

Based on our analysis, the limit of \( \frac{3}{x^4} \) as \( x \) approaches infinity is 0, because the function value diminishes to 0 with increasing \( x \).

Key Concepts

infinity

Infinity is the concept of something that is unbounded or limitless. It is often symbolized by the symbol \( \infty \). In mathematics, when we speak about limits, infinity helps us describe the behavior of functions as inputs grow larger and larger without bound.
- In the expression \( \lim_{x \to \infty} \), we are basically asking, "What happens to our function when \( x \) keeps increasing without stopping?"
- Since infinity isn't a specific number, but rather a concept, it helps us understand how a function behaves rather than what exact value it achieves.
Mathematicians use the idea of approaching infinity to describe situations where values become arbitrarily large, which helps in understanding trends or "end behavior" of functions.
Take for example the expression \( \lim_{x \to \infty} \frac{3}{x^4} \). In this situation, as \( x \) grows larger, \( x^4 \) grows even larger, which means our fraction shrinks towards zero despite its numerator having a constant value.

rational functions

A rational function is any function that can be expressed as the quotient of two polynomials. In more straightforward terms, it's any function that looks like a fraction whose top and bottom are both polynomials.
- The general form is \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
- They are similar to the fractions we are used to in arithmetic, except they involve variables like \( x \).
In our example, \( \frac{3}{x^4} \) is a rational function because the numerator (3) and the denominator (\( x^4 \)) are both simple polynomials.
Rational functions help us understand complex relationships in mathematics because they involve division, which is a common operation.
One common task with rational functions is determining their limits as \( x \) grows very large, which can showcase their end behavior.

behavior of functions at infinity

The behavior of functions at infinity refers to how the functions 'act' or what values they tend to as their inputs become extremely large (or sometimes extremely negative). It's about seeing the long-term trend or attention direction of a function.
- For many functions, especially rational functions, figuring out what happens as \( x \to \infty \) helps us understand key characteristics, like horizontal asymptotes.
- If the function approaches a certain number, we say it has a horizontal asymptote at that value.
Analyzing functions as they approach infinity is critical in calculus because it tells us about the stability or instability of a system, among other things.
In the context of our problem \( \lim_{x \to \infty} \frac{3}{x^4} = 0 \), as \( x \) increases, the term \( \frac{3}{x^4} \) shrinks towards zero, indicating that the function's horizontal asymptote is at \( y = 0 \). This means, no matter how large \( x \) gets, \( \frac{3}{x^4} \) will never actually reach or go below zero but will hover indefinitely closer.