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BITSAT 2013
Exam
BITSAT
Year
2013
Subject
Mathematics
Chapter
Limits

BITSAT 2013 · Mathematics · Limits

Let f(x)= (x^ 5 -1)(x^ 3 +1), g(x)= (x^ 2 -1)(x^ 2 -x+1) and let...

Let⁡f(x)=\quad \operatorname{Let} f(x)=\left(x^{5}-1\right)\left(x^{3}+1\right), g(x)= \left(x^{2}-1\right)\left(x^{2}-x+1\right)andletand let\mathrm{h}(\mathrm{x})besuchthatbe such that\mathrm{f}(\mathrm{x})=\mathrm{g}(\mathrm{x}) \mathrm{h}(\mathrm{x}).ThenThen\operatorname{limh}_{x → 1} \mathrm{ h}(\mathrm{x}) is

Options

  1. A0
  2. B1
  3. C3
  4. D5

Correct answer

D. 5

Step-by-step solution

Given f(x)=g(x)h(x)⇒h(x)=f(x)g(x)⇒lim⁡x→1h(x)=lim⁡x→1f(x)g(x)⇒lim⁡x→1(x5−1)(x3+1)(x2−1)(x2−x+1)=lim⁡x→1x5−15x−1=5×14=5f(x)=g(x) h(x)\Rightarrow h(x)=\frac{f(x)}{g(x)}\Rightarrow \lim _{x \rightarrow 1} h(x)=\lim _{x \rightarrow 1} \frac{f(x)}{g(x)}\Rightarrow \lim _{x \rightarrow 1} \frac{\left(x^{5}-1\right)\left(x^{3}+1\right)}{\left(x^{2}-1\right)\left(x^{2}-x+1\right)}=\lim _{x \rightarrow 1} \frac{x^{5}-1^{5}}{x-1}=5 \times 1^{4}=5
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