Problem 62 - Applied Calculus

Question

\(61-64\). Which of the two limits exists? a. \(\lim _{x \rightarrow-\infty} x^{3}\) b. \(\lim _{x \rightarrow-\infty} \frac{1}{x^{3}}\)

Step-by-Step Solution

Answer

Limit b exists and is equal to 0; limit a does not exist.

Step 1: Analyze limit a

For part a, we have the limit \(\lim_{x \rightarrow -\infty} x^3\). As \(x\) approaches \(-\infty\), \(x\) becomes more and more negative. Since \(x^3\) is a cubic function with a negative leading coefficient when \(x\) is negative, the value of \(x^3\) will continue to decrease. Therefore, \(x^3\) approaches \(-\infty\).

Step 2: Analyze limit b

For part b, we have the limit \(\lim_{x \rightarrow -\infty} \frac{1}{x^3}\). As \(x\) approaches \(-\infty\), \(x^3\) approaches \(-\infty\), making \(\frac{1}{x^3}\) approach 0, since the reciprocal of a very large negative number approaches 0 from the negative side.

Step 3: Conclusion

The limit for part a does not exist because it approaches \(-\infty\). The limit for part b exists and is equal to 0.

Key Concepts

Infinity in Calculus

Infinity is a fascinating concept in calculus, which often confuses students.
In mathematics, infinity behaves a bit differently than regular numbers.
It represents a concept rather than a specific number or value.
It's used to describe things that are unbounded or limitless.
This includes values increasing or decreasing without end. In calculus, when we say something approaches infinity,

• We mean it grows larger and larger without stopping.
• Similarly, approaching negative infinity means something decreases endlessly.

Infinity also helps mathematicians describe the behavior of functions as input values become very large or very small.
When doing calculus problems, like limits, understanding infinity helps us predict how a function behaves at extreme values. Remember:

• Infinity is not a number, but a concept.
• Positive infinity is different from negative infinity.
• Infinity can describe the behavior of functions and their limits.

Limit of a Function

A fundamental idea in calculus is the concept of a limit, which helps us understand a function's behavior.
A limit examines what happens to a function as inputs get close to a certain number.
To grasp limits, picture a function getting continually closer to a specific value without ever actually reaching it.
This potential value is the limit.
We use limits to find out how functions act as inputs get very large, very small, or even approach specific points.
Consider the example of finding

• The limit of a function as inputs approach zero, or
• The direction values take as inputs approach infinity.

Calculating limits involves considering the value that the function tends toward.
In practice, we often write out a limit using notation like this: \(\lim_{x \rightarrow a} f(x)\)This shows that we want to know what happens to the function \(f(x)\) as \(x\) gets very close to \(a\).
With limits, we explore whether the function approaches a specific number, positive infinity, negative infinity, or if it behaves differently.
This helps in analyzing and understanding a function thoroughly.

Asymptotic Behavior

Asymptotic behavior is a key concept when discussing limits and functions in calculus.
This describes what happens to a function as inputs become infinitely large or small.
It helps us predict how functions behave as they approach specific very large or very small values. When we talk about asymptotes, we're discussing lines that a function gets closer to but never actually meets.
The function hugs these lines without ever exactly touching them, no matter how far we extend along the graph.

• Vertical asymptotes occur when a function shoots up indefinitely, unable to reach that line.
• Horizontal asymptotes describe what happens to a function as inputs approach positive or negative infinity.

In our exercise, asking about limits as \(x\) approaches negative infinity involves exploring asymptotic behavior.
We are interested in how the function behaves near these boundaries to better understand function dynamics.
This subject can sometimes be tricky, but it provides crucial insights into the function's behavior at its edges. To grasp this concept:

• Visualize graphs of functions getting closer to lines as inputs go to extremes.
• Realize that asymptotic analysis helps predict function behavior at infinity.