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Question
Quick DL Method Solution
Given:
\lim_{x \to 1} \frac{x^4 - 1}{x - 1} = \lim_{x \to k} \frac{x^3 - k^2}{x^2 - k^2}
LHS using derivative:
\lim_{x \to 1} \frac{x^4 - 1}{x - 1} = \left.\frac{d}{dx}(x^4)\right|_{x=1} = 4x^3|_{x=1} = 4
RHS using DL logic:
\lim_{x \to k} \frac{x^3 - k^2}{x^2 - k^2} \approx \frac{3k^2(x - k)}{2k(x - k)} = \frac{3k}{2}
Equating both sides:
\frac{3k}{2} = 4 \Rightarrow k = \frac{8}{3}
\boxed{k = \frac{8}{3}}
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Commented Feb 10 , 2024
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