Question

Find the limit. $$\lim _{x \rightarrow+\infty}[\ln x-\ln (1+x)]$$

   Find the limit.
$$\lim _{x \rightarrow+\infty}[\ln x-\ln (1+x)]$$

Calculus A New Horizon

Calculus A New Horizon

Howard Anton 6th Edition

Chapter 4, Problem 36

Instant Answer

Solved by Expert Carson Merrill

07/05/2021

Step 1

We can simplify this using the properties of logarithms to get $\lim _{x \rightarrow+\infty}[\ln \frac{x}{1+x}]$. Show more…

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Find the limit. $$\lim _{x \rightarrow+\infty}[\ln x-\ln (1+x)]$$

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Key Concepts

Limit at Infinity

This concept deals with the behavior of functions as the input variable grows without bound. Evaluating limits at infinity helps in understanding the long-term behavior of functions, such as horizontal asymptotes or end behaviors, and is a fundamental part of calculus.

Properties of Logarithms

The properties of logarithms, such as the difference rule (ln(a) - ln(b) = ln(a/b)), are used to simplify logarithmic expressions. These rules are essential in both algebra and calculus to make complex expressions more manageable before applying other operations like limits.

Algebraic Simplification

Algebraic simplification involves manipulating expressions into forms that are easier to evaluate. In calculus, this technique is crucial for revealing the behavior of functions, especially when dealing with limits, as it often resolves indeterminate forms into expressions that can be more directly analyzed.

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