Questions: lim as x approaches 2 from the positive side of 1/(x-2)

Transcript text: $\lim _{x \rightarrow 2^{+}} \frac{1}{x-2}$

Solution

Solution Steps

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.