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GATE 2021 CE (CE1) – Question 28

Engineering Mathematics · Calculus: Limits, continuity, mean value theorems, maxima and minima, Taylor series · 1 mark · Numerical answer

Find $\lim_{x\to1}\left(1/\ln x-1/(x-1)\right)$, correct to one decimal place.

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Correct answer: .5 to .5

Explanation

Put $h=x-1$. Since $\ln(1+h)=h-h^2/2+O(h^3)$, its reciprocal is $1/h+1/2+O(h)$. Subtracting 1/h leaves the limit **0.5**. Evaluating each divergent term separately would lose the cancellation.