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STD 11 - 12. limits — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsSTD 11 - 12. limits1 Mark
MCQ
For $x \in R,\,\,\,\mathop {\lim }\limits_{x \to \infty } \,{\left( {\frac{{x - 3}}{{x + 2}}} \right)^x}$ is equal to
  • A
    $e$
  • B
    ${e^{ - 1}}$
  • ✓
    ${e^{ - 5}}$
  • D
    ${e^5}$
✓

Answer

Correct option: C.
${e^{ - 5}}$
c
(c) $\mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{x + 2 - 5}}{{x + 2}}} \right)^x} = \mathop {\lim }\limits_{x \to \infty } {\left[ {{{\left( {1 - \frac{5}{{x + 2}}} \right)}^{\frac{{x + 2}}{{ - 5}}}}} \right]^{ - \,\frac{{5x}}{{x + 2}}}} = {e^{ - 5}}$ 

$\left( {\because \,\mathop {\lim }\limits_{x \to \infty } \frac{{ - 5x}}{{x + 2}} = \,\mathop {\lim }\limits_{x \to \infty } \frac{{ - 5}}{{1 + \frac{2}{x}}} =  - 5} \right)$.

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