Understanding the Functions

The question provides definitions for three functions: \(f(x)\), \(g(x)\), and \(h(x)\). Let's break them down:

• Absolute Value Function: \(f(x) = |x|\)
• Greatest Integer Function: \(g(x) = [x] - 1\). The notation \([x]\) represents the greatest integer less than or equal to \(x\).
• Composite Function: \(h(x) = \frac{f(g(x))}{g(f(x))}\)

Our goal is to evaluate the right-hand limit of \(h(x)\) as \(x\) approaches 0, denoted as \(\displaystyle \lim _{x \rightarrow 0+}\) \(h(x)\).

Formulating the Function \(h(x)\)

First, let's substitute the definitions of \(f(x)\) and \(g(x)\) into the expression for \(h(x)\):

The numerator is \(f(g(x))\). Since \(f(x) = |x|\), \(f(g(x)) = |g(x)|\). Substituting \(g(x) = [x] - 1\), we get:

\(f(g(x)) = |[x] - 1|\)

The denominator is \(g(f(x))\). Since \(g(x) = [x] - 1\), \(g(f(x)) = [f(x)] - 1\). Substituting \(f(x) = |x|\), we get:

\(g(f(x)) = [|x|] - 1\)

So, the function \(h(x)\) is given by:

Evaluating the Right-Hand Limit as \(x \rightarrow 0+\)

We need to find \(\displaystyle \lim _{x \rightarrow 0+} h(x)\). This means we consider values of \(x\) that are very close to 0 but are slightly greater than 0. For instance, \(x\) could be 0.1, 0.01, 0.001, and so on.

Let's analyze the numerator and denominator separately as \(x \rightarrow 0^+\).

Numerator Analysis: \( |[x] - 1| \) as \(x \rightarrow 0^+\)

As \(x \rightarrow 0^+\), \(x\) is a small positive number. For any \(x\) such that \(0 < x < 1\), the greatest integer less than or equal to \(x\), \([x]\), is 0.

So, for \(x\) close to 0 on the right side, \([x] = 0\).

Therefore, \(|[x] - 1| = |0 - 1| = |-1| = 1\).

Denominator Analysis: \( [|x|] - 1 \) as \(x \rightarrow 0^+\)

As \(x \rightarrow 0^+\), \(x\) is a small positive number. The absolute value of a positive number is the number itself, so \(|x| = x\).

Thus, the denominator becomes \([x] - 1\) for \(x > 0\).

Again, for \(x\) close to 0 on the right side (\(0 < x < 1\)), \([x] = 0\).

Therefore, \( [|x|] - 1 = [x] - 1 = 0 - 1 = -1 \).

Putting it Together for the Limit

As \(x \rightarrow 0^+\), we found that:

• The numerator \(|[x] - 1|\) approaches 1.
• The denominator \( [|x|] - 1 \) approaches -1.

So, the limit of \(h(x)\) as \(x \rightarrow 0^+\) is the limit of the ratio of these values:

Thus, the right-hand limit of \(h(x)\) as \(x \rightarrow 0^+\) is -1.

Conclusion on the Limit

Based on the analysis of the numerator and the denominator of \(h(x)\) as \(x\) approaches 0 from the right side, the limit of \(h(x)\) is -1.

Revision Table - Functions and Limits

Additional Information - Related Concepts

Understanding the behavior of absolute value and greatest integer functions near points where their definition changes (like 0 for |x| and integers for [x]) is crucial for evaluating limits.

• Absolute Value near 0: For small positive \(x\), \(|x|=x\). For small negative \(x\), \(|x|=-x\). This difference is why left-hand and right-hand limits for functions involving \(|x|\) might differ at 0.
• Greatest Integer Function near an Integer: The greatest integer function \([x]\) is discontinuous at every integer.
  • As \(x \rightarrow n^+\) (approaching an integer \(n\) from the right), \([x] = n\) for \(x\) slightly greater than \(n\).
  • As \(x \rightarrow n^-\) (approaching an integer \(n\) from the left), \([x] = n-1\) for \(x\) slightly less than \(n\).
  In this problem, we approached 0 from the right, so \([x] \rightarrow 0\) and \([|x|] \rightarrow 0\) (since \(|x|=x\) for \(x>0\)). If we were evaluating the left-hand limit as \(x \rightarrow 0^-\), \([x]\) would approach -1, and \(|x|\) would be \(-x\) (a small positive number), so \([|x|] = [-x]\) would approach 0. This highlights the importance of the direction of the limit.

Composite functions require evaluating the inner function's behavior first, then the outer function's behavior on that result, especially when dealing with limits near points of discontinuity or points where function definitions change.