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NDA 2018 Maths · Mathematics · Derivatives · easy

The function f(x)=begincases(e^3x-1)/(x),&x≠0\\[2mm] k,&x=0endcases is continuous at x=0 when k equals

Single correct option

The function f(x)={e3x−1x,x≠0k,x=0f(x)=\begin{cases}\dfrac{e^{3x}-1}{x},&x\ne0\\[2mm] k,&x=0\end{cases} is continuous at x=0x=0 when kk equals
  1. A. 00
  2. B. 33
  3. C. 11
  4. D. e3e^3
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Answer: (B) 33

Continuity at 0 requires k=lim⁡x→0e3x−1x=3lim⁡e3x−13x=3⋅1=3k=\lim_{x\to0}\frac{e^{3x}-1}{x}=3\lim\frac{e^{3x}-1}{3x}=3\cdot1=3. The value 1 is the standard limit with no coefficient, and substitution gives the indeterminate form 00\frac00, not 0.
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Original question written in the NDA 2018 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake