Chapter 10: Q. 92 (page 777)

Use L’Hopital’s rule to prove that every power function ˆ with a positive power dominates the logarithmic function $g (x) = \ln x$

Short Answer

Expert verified

Every power functions $x^{n}$ dominate the logarithmic function $\ln x .$

Step by step solution

Step 1. Given information

Exponential growth function and power function are $e^{x}$ and $x^{n}, n \in Z$ respectively.

Step 2. Calculation

Consider,

$\lim_{x \to \infty} \frac{x^{n}}{\ln (x)} (\frac{\infty}{\infty} f o r m); n \in Z^{+}$

Applying Hopital's rule

$= \lim_{x \to \infty} \frac{n x^{n - 1}}{\frac{1}{x}}$

Again Applying Hopital's rule,

$= \lim_{x \to \infty} n x^{n}$

since n is positive

$= \infty \forall x \in [0, 1] = \lim_{x \to \infty} \frac{x^{n}}{\ln (x)} = \infty$

Which is possible only if $x^{n}$ dominates $\ln (x) .$

Thus, every power functions $x^{n}$ dominate the logarithmic function $\ln (x) .$

Hence, theorem proved.

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Use L’Hopital’s rule to prove that every power function ˆ with a positive power dominates the logarithmic function $g (x) = \ln x$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

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Use L’Hopital’s rule to prove that every power function ˆ with a positive power dominates the logarithmic functiong(x)=lnx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Geometry

Calculus

Discrete Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Use L’Hopital’s rule to prove that every power function ˆ with a positive power dominates the logarithmic function $g (x) = \ln x$