(a) Use a CAS to show that if k is a positive constant, thenlimx+x(k1/x 1) = ln k(b)

Chapter 3, Problem 65

(choose chapter or problem)

(a) Use a CAS to show that if k is a positive constant, then

\(\lim _{x \rightarrow+\infty} x\left(k^{1 / x}-1\right)=\ln k\)

(b) Confirm this result using L'Hôpital's rule. [Hint: Express the limit in terms of \(t=1 / x\) ]

(c) If n is a positive integer, then it follows from part (a) with \(x=n\) that the approximation

\(n(\sqrt[n]{k}-1) \approx \ln k\)

should be good when n is large. Use this result and the square root key on a calculator to approximate the values of ln0.3 and ln2 with \(n=1024\), then compare the values obtained with values of the logarithms generated directly from the calculator. [Hint: The nth roots for which n is a power of 2 can be obtained as successive square roots.]

Equation Transcription:

Text Transcription:

lim _x right arrow + infinity x (k^1/x -1)=ln k

t=1/x

x=n

n(nth root k  -1) approx =ln k

n=1024