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12. limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS12. limits1 Mark
MCQ
If $a,\;b,\;c,\;d$ are positive, then $\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{1}{{a + bx}}} \right)^{c + dx}} = $
  • ✓
    ${e^{d/b}}$
  • B
    ${e^{c/a}}$
  • C
    ${e^{(c + d)/(a + b)}}$
  • D
    $e$
✓

Answer

Correct option: A.
${e^{d/b}}$
a
(a) $\mathop {\lim }\limits_{x \to \infty } \,{\left( {1 + \frac{1}{{a + bx}}} \right)^{c + dx}} $

$= \mathop {\lim }\limits_{x \to \infty } \,{\left\{ {{{\left( {1 + \frac{1}{{a + bx}}} \right)}^{a + bx}}} \right\}^{\frac{{c + dx}}{{a + bx}}}} = {e^{d/b}}$

$\left\{ {\because \,\,\mathop {\lim }\limits_{x \to \infty } \,{{\left( {1 + \frac{1}{{a + bx}}} \right)}^{a + bx}} = e} \right.$  and 

$\left. {\mathop {\lim }\limits_{x \to \infty } \frac{{c + dx}}{{a + bx}} = \frac{d}{b}} \right\}$.

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