Problem 46 - Calculus - AP Edition

Question

Show that for any positive integer \(n\) (a) \(\lim _{x \rightarrow+\infty} \frac{\ln x}{x^{n}}=0\) (b) \(\lim _{x \rightarrow+\infty} \frac{x^{n}}{\ln x}=+\infty\).

Step-by-Step Solution

Answer

(a) The limit is 0. (b) The limit is +∞.

Step 1: Set Up the Limit for Part (a)

We want to prove \( \lim_{x \rightarrow +\infty} \frac{\ln x}{x^{n}} = 0 \). Start by setting \( L = \lim_{x \rightarrow +\infty} \frac{\ln x}{x^n} \). To show \( L = 0 \), we will use L'Hôpital's Rule, which requires both the numerator and denominator to approach either \( \infty \) or \( 0 \) as \( x \to \infty \).

Step 2: Apply L'Hôpital's Rule

Both \( \ln x \to \infty \) and \( x^n \to \infty \) as \( x \to \infty \). Therefore, the conditions for L'Hôpital's Rule are satisfied. Differentiate the numerator and the denominator separately:- Derivative of \( \ln x \) is \( \frac{1}{x} \).- Derivative of \( x^n \) is \( nx^{n-1} \).Thus, by L'Hôpital's Rule, the limit becomes:\[ L = \lim_{x \rightarrow +\infty} \frac{1/x}{nx^{n-1}} = \lim_{x \rightarrow +\infty} \frac{1}{nx^n}. \]

Step 3: Evaluate the New Limit for Part (a)

Now, evaluate \( \lim_{x \rightarrow +\infty} \frac{1}{nx^n} \). As \( x \to \infty \), \( x^n \to \infty \), which makes \( \frac{1}{nx^n} \to 0 \). Hence, we conclude that:\( \lim_{x \rightarrow +\infty} \frac{\ln x}{x^n} = 0 \).

Step 4: Set Up the Limit for Part (b)

Now, we need to show that \( \lim_{x \rightarrow +\infty} \frac{x^n}{\ln x} = +\infty \). Consider the reciprocal of the limit in part (a), which can be interpreted as applying L'Hôpital's Rule in reverse.

Step 5: Explore the Reciprocal Limit

From part (a), we know \( \lim_{x \rightarrow +\infty} \frac{\ln x}{x^n} = 0 \), indicating that \( \frac{\ln x}{x^n} \) gets smaller than any positive number as \( x \to \infty \). Thus, its reciprocal \( \frac{x^n}{\ln x} \) must get larger than any positive number as \( x \to \infty \), implying that \( \lim_{x \rightarrow +\infty} \frac{x^n}{\ln x} = +\infty \).

Key Concepts

Understanding Limits at Infinity

When we talk about limits at infinity, we are exploring what happens to a function as its input values become very large, in positive or negative direction. Specifically, when we write \[ \lim_{x \rightarrow +\infty} f(x) = L \],it means that as \( x \) grows without bound, the function \( f(x) \) approaches the value \( L \). In the exercise, we are focusing on examining the behavior of the natural logarithm and polynomial functions as \( x \) heads towards infinity. This helps us understand how different types of growth compare to each other. Essentially, understanding limits at infinity gives insight into the "end behavior" of functions, particularly useful for analyzing functions that model real-world scenarios, like exponential and logarithmic growth. We investigate if a function settles at a particular value, continues to grow or shrink, or even oscillates.

Logarithmic Functions

Logarithmic functions, typically like \( \ln x \) (natural logarithm), grow at a very slow pace compared to other functions like polynomials. It's essential to grasp that:

• As \( x \) increases, \( \ln x \) also increases but at an increasingly slower rate.
• Remember the shape: logarithmic functions have a curve that rises steeply initially, then flattens out.
• Even though \( \ln x \) approaches infinity as \( x \) does, it does so much slower than polynomial functions of any positive degree.

In the context of the exercise, we compare \( \ln x \) with \( x^n \), a polynomial, showing that as \( x \rightarrow +\infty \), \( \ln x \) indeed becomes insignificant compared to any polynomial growth; this supports the use of L'Hôpital's Rule for a clearer view on their relationship.

Polynomial Growth

Polynomial functions, such as \( x^n \), display different growth characteristics based upon their degree, \( n \). Key points include:

• Higher-degree polynomials grow faster than lower-degree ones as \( x \rightarrow +\infty \).
• Any polynomial \( x^n \) grows faster than logarithmic functions like \( \ln x \) as \( x \) becomes very large.

In the exercise, understanding polynomial growth is crucial because it allows us to apply L'Hôpital's Rule effectively. Because polynomials dominate in their growth over logarithms, this shows why the reciprocal functions, like \( \frac{x^n}{\ln x} \), tend to infinity as \( x \) grows. Comprehending the exponential nature of polynomials compared to logarithms forms the foundation for evaluating and proving limits at infinity.