To solve the limit as \( x \) approaches 2 from the right of the function \( \frac{1}{x-2} \), we need to analyze the behavior of the function as \( x \) gets closer to 2 from values greater than 2. As \( x \) approaches 2 from the right, \( x-2 \) approaches 0 from the positive side, making the fraction \( \frac{1}{x-2} \) grow without bound towards positive infinity.
Step 1: Define the Limit Expression
We are given the limit expression:
\[ \lim_{x \to 2^{+}} \frac{1}{x-2} \]
Step 2: Analyze the Behavior as \( x \) Approaches 2 from the Right
As \( x \) approaches 2 from the right (\( x \to 2^{+} \)), the term \( x-2 \) approaches 0 from the positive side. This makes the denominator of the fraction very small and positive.
Step 3: Determine the Limit
Since the denominator \( x-2 \) approaches 0 from the positive side, the fraction \( \frac{1}{x-2} \) grows without bound towards positive infinity.