Question

$$\lim_{x \to 2} (x - 1)^{\frac{1}{3x - 6}} =$$

• A. $e^2$
• B. $e^3$
• C. $e^{\frac{1}{3}}$
• D. $e^{\frac{1}{2}}$

Hint:

Whenever a limit has the form $\lim_{t \to 0} (1 + t)^{\frac{k}{t}}$, its value is simply $e^k$. By rewriting our limit with a substitution $t = x - 2$, the expression becomes $\lim_{t \to 0} (1 + t)^{\frac{1}{3t}}$, which directly gives $e^{\frac{1}{3}}$ in one step!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
Evaluate the limit lim(x→2) (x–1)^(1/(3x–6)), which yields the indeterminate form 1^∞.

Step 2: Key Formula or Approach:
For 1^∞ forms, use L = e^{lim [f(x)–1]·g(x)}.

Step 3: Detailed Explanation:
Here f(x)=x–1, g(x)=1/(3x–6). Exponent = lim (x–2)·1/(3(x–2)) = 1/3. Thus L = e^(1/3).

Step 4: Final Answer:
The limit is e^(1/3), matching option (C).