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

lim as x approaches 2 from the positive side of 1/(x-2)
Transcript text: $\lim _{x \rightarrow 2^{+}} \frac{1}{x-2}$
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Solution

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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.

Final Answer

\(\boxed{\infty}\)

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