Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each limit. Hint: Transform to problems involving a continuous variable \(x\). Assume that \(a>0\). (a) \(\lim _{n \rightarrow \infty} \sqrt[n]{a}\) (b) \(\lim _{n \rightarrow \infty} \sqrt[n]{n}\) (c) \(\lim _{n \rightarrow \infty} n(\sqrt[n]{a}-1)\) (d) \(\lim _{n \rightarrow \infty} n(\sqrt[n]{n}-1)\)

Step-by-Step Solution

Answer

(a) 1, (b) 1, (c) \( \ln(a) \), (d) \( \infty \).

Step 1: Transform Part (a)

For the problem \( \lim_{n \to \infty} \sqrt[n]{a} \), transform it to a function of \( x \). We have \( \sqrt[n]{a} = a^{1/n} \). We are then finding \( \lim_{x \to \infty} a^{1/x} \).

Step 2: Evaluate Limit for Part (a)

Recognize that for any positive \( a \), \( a^{1/x} \rightarrow 1 \) as \( x \rightarrow \infty \) since dividing by \( x \) approaches zero in the exponent, leading to \( a^0 = 1 \). Thus, \( \lim_{n \to \infty} \sqrt[n]{a} = 1 \).

Step 3: Transform Part (b)

For \( \lim_{n \to \infty} \sqrt[n]{n} \), rewrite this as \( (n)^{1/n} \) and recognize the expression resembles \( \lim_{x \to \infty} (x)^{1/x} \).

Step 4: Evaluate Limit for Part (b)

Recognize that \( x^{1/x} \rightarrow 1 \) because the expression can be seen as taking the \( x \)-th root of \( x \), which converges to one. Thus, \( \lim_{n \to \infty} \sqrt[n]{n} = 1 \).

Step 5: Transform Part (c)

For \( \lim_{n \to \infty} n(\sqrt[n]{a} - 1) \), rewrite it as \( n(a^{1/n} - 1) \) and recognize the original variable \( x \) context: \( \lim_{x \to \infty} x(a^{1/x} - 1) \).

Step 6: Simplify the Expression for Part (c)

Use the formula for the expression \( a^{1/x} - 1 \approx \ln(a)/x \) when \( x \) is large by applying L'Hôpital's Rule or expansion theory. Thus, \( x \cdot (a^{1/x} - 1) \approx \ln(a) \). Therefore, \( \lim_{n \to \infty} n(\sqrt[n]{a} - 1) = \ln(a) \).

Step 7: Transform Part (d)

For \( \lim_{n \to \infty} n(\sqrt[n]{n} - 1) \), transform \( n\left((n)^{1/n} - 1\right) \) similarly into \( \lim_{x \to \infty} x((x)^{1/x} - 1) \).

Step 8: Simplify the Expression for Part (d)

Recognize \( (x)^{1/x} \approx 1 + \ln(x)/x \) when \( x \) approaches infinity. Substituting gives \( x((x)^{1/x} - 1) \approx x \cdot \ln(x)/x = \ln(x) \). As \( x \to \infty \), \( \ln(x) \to \infty \), thus \( \lim_{n \to \infty} n(\sqrt[n]{n} - 1) = \infty \).

Key Concepts

Infinite Limits

Understanding the concept of infinite limits is essential in calculus as it deals with the behavior of functions as they approach infinity, or as the input variable grows without bound. An infinite limit can occur when either the function itself grows infinitely large, or when approaching a point where the function becomes unbounded.
Let's consider the exercise you have, especially parts (c) and (d), which involve understanding infinite limits more intuitively.
Often, you will see terms simplify at such limits using mathematical techniques such as

• L'Hôpital's Rule which helps deal with indeterminate forms.
• Logarithmic transformations that simplify the exponential expressions involved.

When examining \(\lim_{n \to \infty} n(\sqrt[n]{n} - 1) \rightarrow \infty\) as seen in step 8, it's indicative of a function's output growing infinitely large. Grasping this concept requires understanding that as \(n\), or \(x\), tends toward infinity, certain patterns will emerge in how the expression behaves.

Continuous Functions

Continuous functions are those where small changes to the input result in small changes to the output. They do not have breaks, jumps, or holes at any given interval. This characteristic makes evaluating limits within continuous functions a straightforward process as limits of continuous functions are simply the value of the function at a point, provided the function is defined there.
In your exercise, especially in parts (a) and (b), the transformations utilize the continuity of functions to simplify the approach to finding these limits.
Consider how \(\lim_{n \to \infty} \sqrt[n]{a} = 1 \). If you track the changes as \(n\) becomes very large, the rooted value approaches smooth behavior due to the continuity, as does \(x^{1/x} \rightarrow 1\),
both simplifying to limit determinations. Whenever you evaluate an expression's limit over a continuous function, you rely on that consistency of behavior.

Exponential Functions

Exponential functions, denoted generally as \(a^x\), where \(a\) is a constant, are essential tools in calculus, especially when dealing with growth processes. Understanding how these functions behave allows us to predict and simplify the behavior of more complex expressions, such as those involving limits.
In the given exercises, transformations of exponential functions help to break down the problems into simpler forms. For example:

• In part (a), \( \sqrt[n]{a} = a^{1/n} \) highlights exponential decay as \(n\) increases.
• In part (c), simplifying \( n(a^{1/n} - 1) \) becomes easier by applying logarithmic transformations.

Many exponential problems in calculus will involve understanding the effects of large or small exponents on such functions. Particularly, recognizing how these functions converge to particular values as variables increase or decrease infinitely is a crucial skill in handling limit problems within this topic.