Problem 57 - Calculus - AP Edition

Question

Limits of the type $$0 / \infty, \quad \infty / 0, \quad 0^{\infty}, \quad \infty \cdot \infty, \quad+\infty+(+\infty),$$ $$+\infty-(-\infty), \quad-\infty+(-\infty), \quad-\infty-(+\infty)$$ are not indeterminate forms. Find the following limits by inspection. (a) \(\lim _{x \rightarrow 0^{+}} \frac{x}{\ln x}\) (b) \(\lim _{x \rightarrow+\infty} \frac{x^{3}}{e^{-x}}\) (c) \(\lim _{x \rightarrow(\pi / 2)^{-}}(\cos x)^{\tan x}\) (d) \(\lim _{x \rightarrow 0^{+}}(\ln x) \cot x\) (e) \(\lim _{x \rightarrow 0^{+}}\left(\frac{1}{x}-\ln x\right)\) (f) \(\lim _{x \rightarrow-\infty}\left(x+x^{3}\right)\)

Step-by-Step Solution

Answer

(a) 0 (b) +∞ (c) 0 (d) -∞ (e) +∞ (f) -∞

Step 1: Analyze the Limit in (a)

To find \(\lim _{x \rightarrow 0^{+}} \frac{x}{\ln x}\), observe that as \(x \rightarrow 0^{+}\), \(x \rightarrow 0\) and \(\ln x \rightarrow -\infty\). Thus, we have \(0 / (-\infty) \rightarrow 0\). The answer is \(0\).

Step 2: Analyze the Limit in (b)

For \(\lim _{x \rightarrow+\infty} \frac{x^{3}}{e^{-x}}\), as \(x \rightarrow +\infty\), \(x^3 \rightarrow +\infty\) and \(e^{-x} \rightarrow 0^{+}\). This gives an \(\infty / 0\) form, which is not indeterminate. The result is \(+\infty\).

Step 3: Analyze the Limit in (c)

To find \(\lim _{x \rightarrow(\pi / 2)^{-}}(\cos x)^{\tan x}\), note that as \(x \rightarrow \pi/2^{-}\), \(\cos x \rightarrow 0^{+}\) and \(\tan x \rightarrow +\infty\). This forms \(0^{+\infty}\), which is \(0\).

Step 4: Analyze the Limit in (d)

For \(\lim _{x \rightarrow 0^{+}}(\ln x) \cot x\), as \(x \rightarrow 0^{+}\), \(\ln x \rightarrow -\infty\) and \(\cot x \rightarrow +\infty\). This results in \(-\infty \times +\infty\), which is \(-\infty\).

Step 5: Analyze the Limit in (e)

For \(\lim _{x \rightarrow 0^{+}}\left(\frac{1}{x}-\ln x\right)\), observe that as \(x \rightarrow 0^{+}\), \(\frac{1}{x} \rightarrow +\infty\) and \(\ln x \rightarrow -\infty\). The form \(+\infty - (-\infty)\) simplifies to \(+\infty\).

Step 6: Analyze the Limit in (f)

To evaluate \(\lim _{x \rightarrow-\infty}\left(x+x^{3}\right)\), consider that \(x \rightarrow -\infty\) means both \(x\) and \(x^3\) (which dominates since \(-\) raised to an odd power is \(-\)) tend towards \(-\infty\). Thus, the limit is \(-\infty\).

Key Concepts

Indeterminate Forms

In calculus, indeterminate forms are expressions that arise when we attempt to evaluate certain types of limits. These forms do not have a clear or obvious value without further analysis. Common examples include expressions like \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \cdot \infty \), \( \infty - \infty \), \( 1^\infty \), \( 0^0 \), and \( \infty^0 \).
Such forms require special techniques to evaluate, like L'Hôpital's Rule, algebraic manipulation, or series expansion. They present an opportunity to explore deeper properties of functions and their behaviors as they approach certain values.
However, it's worth noting that not all forms are indeterminate. For example, \( \frac{0}{\infty} \) and \( \infty \cdot \infty \) are not considered indeterminate because their limits can be determined directly to be \( 0 \) and \( \infty \) respectively, under specific conditions.

Limit Evaluation

Limit evaluation is a critical aspect of calculus that involves finding the value that a function approaches as the input (or variable) approaches a specific point. Finding a limit can sometimes involve straightforward substitution, while other scenarios require more nuanced approaches.
In the original exercise, each limit had to be evaluated by observing the behavior of individual function components as the variable approaches a limit point.

• For example, to evaluate \( \lim _{x \rightarrow 0^{+}} \frac{x}{\ln x} \), a direct substitution wouldn't work because it leads to the form \( 0/(-\infty) \). However, understanding that \( \ln x \to -\infty \) allows us to conclude that the limit approaches \( 0 \).
• Each limit requires identifying key components and considering their limits to derive the overall limit. Simple forms like \( \frac{0}{\infty} \) do not need further calculation as they can be evaluated more directly as 0.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as the input approaches a particular point, often infinity or some finite value. This behavior is crucial in evaluating limits, especially when the output is not readily apparent.
For instance, in evaluating \( \lim _{x \rightarrow +\infty} \frac{x^{3}}{e^{-x}} \), as \( x \) increases, \( x^3 \) grows indefinitely, and \( e^{-x} \) approaches zero. The result is an \( \infty/0 \) form, where the numerator dominates the denominator, leading to a limit of \( +\infty \).
Recognizing asymptotic behavior is essential in identifying when certain terms become negligible and others dominant. This helps simplify complex relationships to evaluate limits effectively.

Infinity in Calculus

Infinity is a concept in calculus that describes unbounded behavior or quantities. It is not a number but an idea that represents endless continuation. Understanding how functions behave as they approach infinity is essential because it helps describe their long-term behavior.
In the original exercise, several limits involved terms approaching \( +\infty \) or \( -\infty \). For instance:

• \( \lim _{x \rightarrow -\infty}(x+x^{3}) \): Here, \( x^3 \) becomes very large and negative much faster than \( x \), so the expression tends towards \( -\infty \).
• Situations like \( +\infty - (-\infty) \) simplify to \( +\infty \) because subtracting a negative infinity from a positive infinity results in something infinitely large and positive.

Interactions involving infinity often simplify complex expressions in limits, helping describe how functions behave at extreme values.