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Limits — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits5 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{9}-\text{a}^9}{\text{x}-\text{a}}=9,$ find all possible value of a.
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Answer

$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{9}-\text{a}^9}{\text{x}-\text{a}}=9\ \cdots{\text{(i)}}$
$\text{L.H.S}=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{9}-\text{a}^9}{\text{x}-\text{a}}$
$=9(\text{a})^{9-1}$
$=9\text{a}^{8}$
It is given that $9^\text{a}=9$ [From (i)]
$\Rightarrow\text{a}^{8}=\frac99=1$
$\Rightarrow\text{a}^4=1$
$\text{a}^2=1$
$\text{a}=\pm1$
$\Rightarrow \text{a} = 1 \text{ and a} = -1$

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