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Chapter 1: Problem 44

\(\lim _{x \rightarrow 0^{-}}(\ln (\\{x\\}+|[x]|))^{|x\rangle}\) is equal to (A) 0 (B) 1 (C) \(\ln 2\) (D) \(\ln \frac{1}{2}\) where [] is the greatest integer function and \\{\\} is the fractional part function.

Short Answer

Expert verified
Answer: The limit as x approaches 0 from the negative side of the function \((\ln (\{x\} + |x|))^{|x|}\) is 1 (B).

Step by step solution

01

Understand the Greatest Integer Function and Fractional Part Function

The greatest integer function, denoted by [x], is the largest integer less than or equal to x. The fractional part function, denoted by \{x\}, is the fractional part of x, defined as x - [x].
02

Find Expression for \{x\} and |x| as x Approaches 0 from Left

As x approaches 0 from the left (denoted as 0-), x will be negative. For a negative x, [x] will be the largest integer less than x, which is -1, and therefore \{x\} will be x - [x] = x - (-1) = x + 1. Also, for negative x, the absolute value of x, |x|, is equal to -x.
03

Substitute the Expressions for \{x\} and |x| in the Given Function

Substitute the expressions for \{x\} and |x| in the given function: \((\ln (\{x\} + |x|))^{|x|} = (\ln (x + 1 - x))^{-x} = (\ln (1))^{-x}\)
04

Evaluate the Natural Logarithm of 1

The natural logarithm of 1, \(\ln(1)\), is equal to 0.
05

Substitute the Value of the Natural Logarithm in the Function

Substitute the value of the natural logarithm in the function: \((\ln (1))^{-x} = 0^{-x}\)
06

Evaluate the Limit of the Function as x Approaches 0 from Left

As x approaches 0 from the left, the exponent -x approaches 0 from the right. The limit of \(0^{-x}\) is 1 as x approaches 0 from the right. Therefore, the given limit is equal to: \(\lim _{x \rightarrow 0^{-}}(\ln (\{x\}+|[x]|))^{|x|} = 1\) The answer is (B) 1.

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Most popular questions from this chapter

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