Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits2 Marks
MCQ
If $\lim _{x \rightarrow a} \frac{ a ^x-x^a}{x^x- a ^x}=-1$, then
✓
$a=1$
B
$a =0$
C
$a = e$
D
none of these
✓
Answer
Correct option: A.
$a=1$
(A) $\lim _{x \rightarrow a} \frac{a^x-x^a}{x^x-a^a}=-1$ Applying L-Hospital's rule on L.H.S., we get $\lim _{x \rightarrow a} \frac{a^x \log _e a-a x^{a-1}}{x^x\left(1+\log _e x\right)}=-1$ $\Rightarrow \frac{a^a \log _e a-a \cdot a^{a-1}}{a^a\left(1+\log _e a\right)}=-1$ $\Rightarrow \frac{\log _{ e } a -1}{\log _{ e } a +1}=-1$ $\Rightarrow \log _{ e } a -1=-\log _{ e } a -1$ $\Rightarrow 2 \log _{ e } a =0 \Rightarrow a = e ^0=1$
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