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12. limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS12. limits1 Mark
MCQ
The value of $\mathop {\lim }\limits_{x \to a} \frac{{\log (x - a)}}{{\log ({e^x} - {e^a})}}$ is
  • ✓
    $1$
  • B
    $-1$
  • C
    $0$
  • D
    None of these
✓

Answer

Correct option: A.
$1$
a
(a) $\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to a} \,\,\frac{{\log \,(x - a)}}{{\log \,({e^x} - {e^a})}} = \mathop {{\rm{lim}}}\limits_{{\rm{x}} \to a} \,\,\frac{{{e^x} - {e^a}}}{{(x - a)\,{e^x}}}$, $\left( {{\rm{Form}} \,\, \frac{0}{0}} \right)$

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

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