Understanding the Problem: Limit of a Composite Function Ratio

The problem asks us to find the left-hand limit of a function \(h(x)\) as \(x\) approaches 0. The function \(h(x)\) is defined as the ratio of two composite functions: \(f(g(x))\) and \(g(f(x))\). We are given the definitions of \(f(x)\) and \(g(x)\):

• \(f(x) = |x|\) (absolute value function)
• \(g(x) = [x] - 1\) (greatest integer function minus 1)

We need to evaluate \(\displaystyle \lim _{x \rightarrow 0-} h(x) = \lim _{x \rightarrow 0-} \frac{f(g(x))}{g(f(x))}\).

Step-by-Step Calculation of the Left-Hand Limit

To find the limit of \(h(x)\) as \(x \rightarrow 0-\), we need to evaluate the limits of the numerator \(f(g(x))\) and the denominator \(g(f(x))\) separately as \(x\) approaches 0 from the left side.

1. Evaluate the Denominator: Finding \( \lim_{x \rightarrow 0-} g(f(x)) \)

First, let's analyze the inner function \(f(x)\) as \(x \rightarrow 0-\).

• As \(x \rightarrow 0-\), \(x\) takes values slightly less than 0 (e.g., -0.1, -0.01, -0.001, ...).
• \(f(x) = |x|\). For these negative values of \(x\), \(|x| = -x\).
• As \(x \rightarrow 0-\), \(-x\) approaches 0 from the positive side (e.g., -(-0.1) = 0.1, -(-0.01) = 0.01, ...). So, \(f(x) \rightarrow 0+\).

Now, let's evaluate the outer function \(g(y) = [y] - 1\) as its input \(y = f(x)\) approaches 0 from the positive side (\(y \rightarrow 0+\)).

• As \(y \rightarrow 0+\), \(y\) takes values slightly greater than 0 (e.g., 0.1, 0.01, 0.001, ...).
• The greatest integer less than or equal to \(y\), \([y]\), for these values is 0.
• Therefore, as \(y \rightarrow 0+\), \(g(y) = [y] - 1\) approaches \(0 - 1 = -1\).

So, we have \(\displaystyle \lim_{x \rightarrow 0-} g(f(x)) = -1\).

2. Evaluate the Numerator: Finding \( \lim_{x \rightarrow 0-} f(g(x)) \)

Next, let's analyze the inner function \(g(x)\) as \(x \rightarrow 0-\).

• As \(x \rightarrow 0-\), \(x\) takes values slightly less than 0.
• \(g(x) = [x] - 1\). For values of \(x\) in the interval \([-1, 0)\), the greatest integer \([x]\) is -1.
• Therefore, as \(x \rightarrow 0-\), \(g(x)\) takes the value \(-1 - 1 = -2\). Note that \(g(x)\) is constant and equal to -2 as \(x\) approaches 0 from the left (for \(x \in [-1, 0)\)).

Now, let's evaluate the outer function \(f(y) = |y|\) as its input \(y = g(x)\) approaches -2.

• As \(x \rightarrow 0-\), \(g(x)\) is equal to -2.
• Therefore, \(f(g(x))\) is equal to \(f(-2) = |-2|\).
• The absolute value of -2 is 2.

So, we have \(\displaystyle \lim_{x \rightarrow 0-} f(g(x)) = 2\).

3. Calculate the Limit of h(x)

Now we can find the limit of \(h(x)\) using the limits of the numerator and the denominator:

\[ \lim _{x \rightarrow 0-} h(x) = \lim _{x \rightarrow 0-} \frac{f(g(x))}{g(f(x))} = \frac{\lim _{x \rightarrow 0-} f(g(x))}{\lim _{x \rightarrow 0-} g(f(x))} \] \[ \lim _{x \rightarrow 0-} h(x) = \frac{2}{-1} = -2 \]

The left-hand limit of \(h(x)\) as \(x\) approaches 0 is -2.

Final Answer Summary

Based on our analysis, as \(x \rightarrow 0-\), \(f(g(x))\) approaches 2 and \(g(f(x))\) approaches -1. Therefore, the limit of their ratio \(h(x)\) is 2 / (-1) = -2.

Revision Table: Key Concepts in Limit Calculation

Additional Information: Behavior of Functions near Zero

Understanding how \(|x|\) and \([x]\) behave near critical points like 0 is crucial for evaluating limits involving these functions.

• Absolute Value near 0:
  • As \(x \rightarrow 0+\), \(x > 0\), so \(|x| = x \rightarrow 0+\).
  • As \(x \rightarrow 0-\), \(x < 0\), so \(|x| = -x \rightarrow 0+\).
  • The limit as \(x \rightarrow 0\) is 0, and the function approaches 0 from the positive side regardless of whether \(x\) approaches from the left or right.
• Greatest Integer Function near 0:
  • As \(x \rightarrow 0+\), \(0 < x < 1\), so \([x] = 0\). The limit is 0.
  • As \(x \rightarrow 0-\), \(-1 \le x < 0\), so \([x] = -1\). The limit is -1.
  • The function has a jump discontinuity at \(x=0\). The left-hand limit is -1, and the right-hand limit is 0.

In this specific problem, evaluating \(g(f(x))\) as \(x \rightarrow 0-\) means \(f(x) \rightarrow 0+\), so we look at the behavior of \(g\) as its input approaches 0 from the positive side, which gives \([0+] - 1 = 0 - 1 = -1\). Evaluating \(f(g(x))\) as \(x \rightarrow 0-\) means \(g(x)\) is exactly -2 (for \(x \in [-1, 0)\)), so we evaluate \(f\) at -2, which is \(|-2|=2\).