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Chapter 1: Problem 19

Find the limits using your understanding of the end behavior of each function. $$\lim _{x \rightarrow \infty} x^{3}$$

Short Answer

Expert verified
The limit is infinity.

Step by step solution

01

Identify the Function

The function in question is \( x^3 \). This is a polynomial function with a degree of 3.
02

Understand the End Behavior of Polynomials

For polynomials, the end behavior is determined by the leading term. In this case, the leading term is \( x^3 \). For large values of \( x \), the term \( x^3 \) will dominate the behavior of the function.
03

Evaluate the Limit as x approaches Infinity

As \( x \rightarrow \infty \), the term \( x^3 \) will increase without bound. Thus, \( \lim_{x \rightarrow \infty} x^3 = \infty \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

End Behavior of Polynomials
Polynomial functions, like the one in our example, have a unique property known as their "end behavior." To determine what a polynomial does as it stretches towards infinity or plunges towards negative infinity, we need to look at its leading term. The leading term is the term with the highest power of the variable. It gives us a clue about how the polynomial behaves at its "ends."
  • For our polynomial function, the leading term is \(x^3\).
  • This term must be observed as \(x\) grows very large, i.e., as \(x \rightarrow \infty\).
  • The power 3 suggests a positive outcome for positive inputs. As \(x\) increases, \(x^3\) becomes significantly large in the positive direction.
In simple terms, whenever \(x\) is very big and positive, \(x^3\) just keeps getting bigger and bigger in the positive direction.
Polynomial Functions
Polynomial functions are mathematical expressions made up of terms with variables raised to whole number powers, and they can include constant terms. Each term can be a coefficient and a power of a variable like \(x\).
For example, in our exercise, the function is \(x^3\), which is the simplest kind of polynomial called a monomial.
  • Polynomials are classified by their degree, which is determined by the highest power of the variable in the function.
  • The function we've examined here, \(x^3\), is known as a cubic polynomial because the highest power is 3.
  • Cubic polynomials have up to three roots and exhibit various types of curves, depending on their coefficients.
Polynomial functions are central to a number of mathematical investigations because they can beautifully describe both simple curves and complex shapes.
Infinite Limits
The concept of infinite limits involves calculating the behavior of a function as the independent variable approaches an extremely large value, either positive or negative.
When we talk about finding the limit of a function as it approaches positive infinity, like \(\lim_{x \rightarrow \infty} x^3\), we are essentially asking, "What does the function's value trend towards as \(x\) gets infinitely large?"
  • In our case, as \(x\) becomes exceedingly large, \(x^3\) also grows without any bounds or restrictions.
  • This indicates that the function tends towards infinity鈥攚hich is what we call an infinite limit.
  • Infinite limits signify that the function doesn't settle on any fixed value, it simply increases or decreases indefinitely.
Understanding infinite limits is crucial for exploring the asymptotic behavior of functions, allowing us to predict trends and behaviors of polynomials and other mathematical expressions as they approach large-valued limits.

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Most popular questions from this chapter

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