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Chapter 5: Problem 7

Find the limits. $$ \lim _{x \rightarrow-\infty} \sqrt[3]{x} $$

Short Answer

Expert verified
The limit is \(-\infty\).

Step by step solution

01

Understand the Function

The function we need to find the limit for as \(x\) approaches \(-\infty\) is \( \sqrt[3]{x} \), which is the cube root of \(x\). The cube root function \( \sqrt[3]{x} \) is defined for all real numbers, including negative ones.
02

Analyze the Behavior as \(x\) Approaches \(-\infty\)

When \(x\) is a very large negative number, the cube root \(\sqrt[3]{x}\) will also be a negative number because the cube of any negative number is negative. As \(x\) becomes more negative, \( \sqrt[3]{x} \) will become more negative as well.
03

Determine the Limit

Since as \(x\) approaches \(-\infty\), \( \sqrt[3]{x} \) continues to decrease and does not have any boundary, the limit \( \lim_{x \rightarrow -\infty} \sqrt[3]{x} \) is also \(-\infty\).
04

Conclude the Solution

Thus, we conclude that the limit of \( \sqrt[3]{x} \) as \(x\) approaches \(-\infty\) is \(-\infty\).

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Key Concepts

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

Cube Root Function
The cube root function, denoted as \( \sqrt[3]{x} \), is a mathematical operation that returns a number whose cube is \( x \). Unlike the square root function, cube roots operate over both positive and negative numbers without restriction. This is because cubing a negative number results in a negative number, making cube roots always defined for real numbers.
Cube root functions exhibit interesting characteristics:
  • They are continuous and defined for all real numbers \( x \).
  • For positive \( x \), both roots are positive, and for non-positive \( x \), the root is negative or zero.
  • This function is odd, meaning \( \sqrt[3]{-x} = -\sqrt[3]{x} \), which highlights its symmetry around the origin.
Knowing these properties helps in understanding how the function behaves as it approaches negative or positive infinity.
Infinite Limits
Infinite limits are a fundamental concept in calculus that describe the behavior of a function as the input, often represented as \( x \), tends toward infinity or negative infinity. In the context of our example, \( \lim_{x \rightarrow -\infty} \sqrt[3]{x} \), we're interested in how \( \sqrt[3]{x} \) behaves as \( x \) decreases without bound.
Here's why understanding infinite limits is crucial:
  • They help to describe the end behavior of functions, providing insight into function growth or decay.
  • Understanding this concept is essential for identifying asymptotes and determining the long-term trends of functions.
  • Specific to cube root functions, because they can take any real number, the result as \( x \) approaches negative infinity is a negative infinity.
This result reflects the unbounded nature of cube root functions.
Asymptotic Behavior
Asymptotic behavior analyzes how a function behaves as its input grows really large (positively or negatively). For our exercise, we're exploring \( \lim_{x \rightarrow -\infty} \sqrt[3]{x} \), looking at its trend as \( x \) becomes infinitely negative.
This concept is tightly connected to limits and brings critical insights:
  • An asymptote is a line that the graph of a function can approach but never actually reach.
  • In our case, the cube root function doesn't have horizontal asymptotes since it doesn't level off or flatten out.
  • Instead, its asymptotic behavior shows a continual decrease as \( x \) heads toward \(-\infty\).
With this knowledge, you can visualize how functions like the cube root extend across both ends of the real-number spectrum.

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Most popular questions from this chapter

Autocatalytic Reaction The rate \(V\) of a certain autocatalytic reaction is proportional to the amount \(x\) of the product and also proportional to the remaining substance being catalyzed. Thus, if \(I\) is the initial amount of the substance, then \(V=k x(I-x),\) where \(k\) is a positive constant. \({ }^{18}\) At what concentration \(x\) will the rate be maximum?

In Section 4.1 we indicated how to use the limit definition of derivative to find the derivative of \(x^{n},\) where \(n\) is a positive integer, and said that a proof for general \(n\) was somewhat difficult. You can now construct a proof in the case that \(r\) is a rational number. Let \(y=f(x)=x^{p / q}\) where \(p\) and \(q\) are positive integers. Then \(y^{q}=x^{p} .\) Differentiate and show that $$ \frac{d y}{d x}=\frac{p}{q} x^{(p / q)-1} $$

Differentiate implicitly and find the slope of the curve at the indicated point. $$ x y^{2}+x y+y^{3}=5,(2,1) $$

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Give conditions on the constants \(b, c,\) and \(d\) so that the polynomial function \(f(x)=x^{3}+b x^{2}+c x+d\) will be increasing on the interval \((-\infty, \infty)\).
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