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BITSAT 2010
Exam
BITSAT
Year
2010
Subject
Mathematics
Chapter
Continuity and Differentiability

BITSAT 2010 · Mathematics · Continuity and Differentiability

Let (f(x)= ((e^x-1)^2)/( ((x)/(a)) (1+(x)/(4))) ) for x ≠ 0 , and...

Let (f(x)=(ex−1)2sin⁡(xa)log⁡(1+x4)\frac{\left(e^x-1\right)^2}{\sin \left(\frac{x}{a}\right) \log \left(1+\frac{x}{4}\right)}) for x≠0x ≠ 0, and f(0)=12f(0)=12. If (f\mathrm{f}) is continuous at x=0x=0, then the value of aa is equal to

Options

  1. A1
  2. B-1
  3. C2
  4. D3

Correct answer

D. 3

Step-by-step solution

Lt⁡x→0(ex−1)2sin⁡(xa)log⁡(1+x4)=Lt⁡x→0(ex−1)2x⋅x2xa⋅sin⁡(xa)(xa)⋅log⁡(1+x4)x4⋅x4⇒4a=12⇒a=3\begin{aligned}& \operatorname{Lt}_{x → 0} \frac{\left(e^x-1\right)^2}{\sin \left(\frac{x}{a}\right) \log \left(1+\frac{x}{4}\right)}\\ &=\operatorname{Lt}_{x → 0} \frac{\frac{\left(e^x-1\right)^2}{x} \cdot x²}{\frac{x}{a} \cdot \frac{\sin \left(\frac{x}{a}\right)}{\left(\frac{x}{a}\right)} \cdot \frac{\log \left(1+\frac{x}{4}\right)}{\frac{x}{4}} \cdot \frac{x}{4}} \Rightarrow4 a=12 \\ & \Rightarrow a=3 \end{aligned})
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