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GATE 2017 EE – Question 12

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy integral formula, Taylor series, Laurent series, Residue theorem · 1 mark · Multiple choice

For a complex number $z$, $\lim_{z \to i} \dfrac{z^2 + 1}{z^3 + 2z - i(z^2 + 2)}$ is

  1. $-2i$
  2. $-i$
  3. $i$
  4. $2i$

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Show answer and explanation

Correct answer: (D) $2i$

Explanation

At $z = i$ both the numerator and the denominator are 0, so apply L'Hopital's rule. The derivative of the numerator is $2z = 2i$. The derivative of the denominator is $3z^2 + 2 - 2iz$, which at $z = i$ is $-3 + 2 + 2 = 1$. So the limit is $\frac{2i}{1} = 2i$.