Problem 105 a. Assuming that \(\lim _{n \rig... [FREE SOLUTION]

Chapter 10: Problem 105

a. Assuming that $\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0$ if $c$ is any positive constant, show that $$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{t}}=0$$ if $c$ is any positive constant. b. Prove that $\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0$ if $c$ is any positive constant. (Hint: If $\epsilon=0.001$ and $c=0.04,$ how large should $N$ be to ensure that $\left.\left|1 / n^{c}-0\right|<\epsilon \text { if } n>N ?\right)$

Short Answer

Expert verified

a. The limit is 0 by L'H么pital's Rule. b. $\frac{1}{n^c}\to 0$ because $n^c\to \infty$.

Step by step solution

Explore Part a of the Exercise

For part a, we need to show that $\lim _{n \rightarrow \infty} \frac{\ln n}{n^{t}}=0$ for any positive $t$. Recognize that $\ln n$ grows slower than any polynomial power of $n$. Therefore, consider expressing $\ln n$ in terms of $n^{t}$.

Apply L'H么pital's Rule for Part a

Since both the numerator $\ln n$ and the denominator $n^{t}$ approach infinity as $n$ approaches infinity, apply L'H么pital's Rule to find the limit: differentiate the numerator and the denominator. The derivative of $\ln n$ is $1/n$, and the derivative of $n^{t}$ is $tn^{t-1}$.

Evaluate the New Limit for Part a

Now evaluate the new limit: $\lim _{n \rightarrow \infty} \frac{1/n}{tn^{t-1}} = \lim _{n \rightarrow \infty} \frac{1}{tn^{t}} = 0$. This shows that $\ln n$ divided by a polynomial power $n^{t}$ indeed goes to zero as $n$ approaches infinity.

Explore Part b of the Exercise

For part b, we need to show $\lim _{n \rightarrow \infty} \frac{1}{n^{c}} =0$. Assume $\epsilon = 0.001$ and $c = 0.04$, and find $N$ such that if $n > N$, then $\left| \frac{1}{n^{c}} - 0 \right| < \epsilon$.

Solve the Inequality for Part b

To ensure $\frac{1}{n^{c}} < \epsilon$, solve $\frac{1}{n^{0.04}} < 0.001$. This implies $n^{0.04} > 1000$.

Solve for N using Exponents

To find $N$, solve the inequality $n > (1000)^{1/0.04}$. Calculating $ (1000)^{25} $ gives $N \approx 10^{75}$, suggesting a very large value for $N$.

Conclusion and Verification

Thus, for any positive $c$, $\frac{1}{n^c}$ tends to zero as $n$ increases, which confirms the limit assertion for any $c > 0$. This completes both parts of the exercise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

L'H么pital's Rule

When dealing with limits in calculus, especially those involving indeterminate forms like $ \frac{\infty}{\infty} $ or $ \frac{0}{0} $, we have a powerful tool called L'H么pital's Rule. This rule simplifies the process of finding limits by allowing us to differentiate the numerator and denominator until the indeterminate form is resolved.

Here's how it works:

Identify that the function is of the form $ \frac{f(x)}{g(x)} $ and both $ f(x) \to \infty $ and $ g(x) \to \infty $ as $ x \to c $.
Differentiating the numerator $ f(x) $ gives $ f'(x) $, and differentiating the denominator $ g(x) $ gives $ g'(x) $.
Apply the rule: the limit is $ \lim_{x \to c} \frac{f'(x)}{g'(x)} $.

For example, we used L'H么pital's Rule in the exercise to evaluate $ \lim_{n \to \infty} \frac{\ln n}{n^t} $. Differentiating $ \ln n $ gives $ \frac{1}{n} $ and differentiating $ n^t $ results in $ t n^{t-1} $. This application helps us demonstrate that the original limit is zero.

Logarithmic Functions

Logarithmic functions are a crucial element in calculus, especially when dealing with growth rates and asymptotic behavior. The function $ \ln x $ (natural logarithm) is particularly important. It grows at a slower rate compared to polynomial functions, which is key to understanding why in our exercise,$ \lim_{n \to \infty} \frac{\ln n}{n^t} = 0 $.

Key characteristics of $ \ln x $:

The natural logarithm, $ \ln x $, is the inverse of the exponential function $ e^x $.
It is defined for $ x > 0 $ and increases infinitely, but at a decreasing rate.
Unlike polynomial growth $ x^t $, $ \ln x $ grows much slower. This means that no matter how large $ \ln x $ becomes, for any positive $ t $, it will always grow slower than $ n^t $.

Understanding the slow growth rate of logarithmic functions is essential for evaluating many limits involving asymptotic behavior.

Polynomial Functions

Polynomial functions are expressions involving powers of a variable, such as $ x^2 $ or $ x^3 $. They often appear in calculus and have straightforward yet significant properties.

Characteristics of polynomial functions:

The behavior of a polynomial is largely determined by its highest-degree term. For instance, in $ x^3 + 2x^2 + 3x + 4 $, $ x^3 $ is the dominant term as $ x \to \infty $.
Polynomial functions grow without bound as $ x $ increases, if the highest-degree term is positive.
They are continuous and differentiable, making them easier to handle using calculus tools like limits and differentiation.

In the exercise, for any positive constant $ t $, $ n^t $ represents a polynomial function. The contrast between the growth rates of $ \ln n $ and $ n^t $ illustrates why $ \ln n $ becomes negligible compared to $ n^t $ as $ n \to \infty $.

Infinity in Calculus

Infinity is a concept rather than a number. In calculus, we often talk about limits approaching infinity. This notion helps us describe behaviors of functions as variable values become extremely large or small.

Key points about infinity in calculus:

Infinity, $ \infty $, can imply growth without bound when talking about functions like polynomials.
We often encounter $ \infty $ in limits, such as $ \lim_{x \to \infty} f(x) $, describing how functions behave as $ x \to \infty $.
In our exercise, showing $ \lim_{n \to \infty} \frac{1}{n^c} = 0 $ helps outline how fractions involving powers of $ n $ diminish as $ n $ grows.

This idea of infinity allows us to discuss and resolve questions regarding the fate of functions at their extreme bounds, shedding light on fundamental calculus concepts.

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Short Answer

Step by step solution

Explore Part a of the Exercise

Apply L'H么pital's Rule for Part a

Evaluate the New Limit for Part a

Explore Part b of the Exercise

Solve the Inequality for Part b

Solve for N using Exponents

Conclusion and Verification

Key Concepts

L'H么pital's Rule

Logarithmic Functions

Polynomial Functions

Infinity in Calculus

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