Problem 61 - 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

The limit in part (b) exists and equals 0.

Step 1: Evaluate the limit for part (a)

We are asked to find \( \lim _{x \rightarrow \infty} x^{3} \). As \( x \) tends towards infinity, the value of \( x^{3} \) (which is \( x \) multiplied by itself three times) will grow larger and larger without bound. Thus, \( \lim _{x \rightarrow \infty} x^{3} = \infty \), indicating that the limit does not exist because it tends towards infinity.

Step 2: Evaluate the limit for part (b)

Now, let's evaluate \( \lim _{x \rightarrow \infty} \frac{1}{x^{3}} \). As \( x \) increases towards infinity, \( x^{3} \) also becomes infinitely large. Consequently, \( \frac{1}{x^{3}} \) becomes an infinitely small positive number, approaching 0. Therefore, \( \lim _{x \rightarrow \infty} \frac{1}{x^{3}} = 0 \), indicating that this limit exists.

Step 3: Conclusion

After evaluating both limits, we see that the limit for part (a) does not exist as it tends to infinity, while the limit for part (b) exists and equals 0.

Key Concepts

Infinite Limits

When discussing limits, particularly those that involve infinity, we encounter the concept of **infinite limits**. This occurs when the values of a function grow indefinitely large or small as the input approaches a particular value. In other words, as we push the input (\( x \)) toward an infinite boundary, the function tends toward unbounded behavior.
- For example, in the case of \( \lim _{x \rightarrow \infty} x^{3} \), the input \( x \) grows infinitely, causing \( x^{3} \) to explode,unbounded in growth. This means the function's behavior is undefined in a traditional sense, thus we say the limit is \( \infty \) and therefore does not exist in finite terms.
- It's crucial to note that indicating a limit as \( \infty \)is informative of behavior, highlighting how functions could stretch indefinitely, rather than converging on a fixed number.

Limit Evaluation

**Limit evaluation** is the process of determining the specific behavior or outcome of a function as its input approaches a set value. Limits help us understand the trend of a function as it comes close to this point, even if it never reaches it.
In the original task:- For the limit \( \lim _{x \rightarrow \infty} x^{3} \),evaluation reveals an outcome of \( \infty \), as previously described.- In part (b), limit evaluation goes further by assessing\( \lim _{x \rightarrow \infty} \frac{1}{x^{3}} \). As \( x \) becomes larger, \( x^{3} \) grows faster, making\( \frac{1}{x^{3}} \) decrease towards zero, confirming that this limit comfortably exists as 0.

Asymptotic Behavior

Understanding a function's **asymptotic behavior** means analyzing its trend as it approaches a very large value or heads towards infinity. This reveals where and how a function stabilizes or diverges as its input grows.
In many instances:- As with\( \frac{1}{x^{3}} \), an \( x \)-value going toward infinity signifies that the function nears 0. This behavior marks a horizontal asymptote, as the curve infinitely approaches, but never quite reaches, the x-axis.
Recognizing asymptotic tendencies reveals crucial insights about overall function behavior, allowing for predictions and understanding without needing to plot every imaginable value.