limx →0^-(ln({x}+|[x]|))^|x⟩ is equal to (A) 0 (B) 1 (C) ln2 (D) ln(1)/(2) where [] is the great

step 1 Hello everyone from the Gebegmoly we can write that limit. Let us take the limit BL, limit X ten

Limx →0^-(ln({x}+|[x]|))^|x⟩ is equal to (A) 0 (B) 1 (C) ln2...

$\lim _{x \rightarrow 0^{-}}(\ln (\{x\}+|[x]|))^{|x\rangle}$ 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.

Key Concepts

Limits and One?Sided Limits

Limits describe the behavior of functions as the input approaches a particular value. In problems involving one?sided limits, such as approaching zero from the left, it is crucial to consider the particular behavior of components of the function for values only on that one side, which may differ from the behavior on the other side.

Greatest Integer (Floor) Function

The greatest integer function, often denoted by [x], assigns to each real number the largest integer less than or equal to that number. In limit problems, especially when dealing with negative numbers, understanding how the function steps or ‘jumps’ is essential because the value can change abruptly as the variable approaches an integer from below.

Fractional Part Function

The fractional part function, represented by {x}, gives the difference between a real number and the greatest integer less than or equal to that number. When x is negative, its behavior is important because the fractional part will be equal to x minus the floor of x, resulting in values that differ from the typical case for positive x.

Logarithmic Function

The logarithmic function, ln(x), is the inverse of the exponential function and is defined only for positive arguments. In limit evaluation, ensuring that the argument of the logarithm remains within its domain is crucial, and understanding its continuity and behavior near specific points helps in simplifying complex expressions.

Exponential Limit Evaluation

When a function is raised to an exponent that itself is a function of the variable, the evaluation of the limit may involve rewriting the expression using logarithms or employing limit laws such as the fact that any positive number raised to the power approaching zero tends to 1. This concept is essential in resolving limits where the base approaches a constant and the exponent tends to zero.

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