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Limits — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits5 Marks
Question
Evaluate the following limit:
If $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{9}-\text{a}^9}{\text{x}-\text{a}}=\lim\limits_{\text{x}\rightarrow5}(4+\text{x}),$ find all possible value of a.
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Answer

If $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{9}-\text{a}^9}{\text{x}-\text{a}}=\lim\limits_{\text{x}\rightarrow5}(4+\text{x})\ \cdots{\text{(i})}$
$\text{L.H.S}=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{9}-\text{a}^9}{\text{x}-\text{a}}$
$=9(\text{a})^{9-1}$
$=9\text{a}^{8}\ \cdots{\text{(ii})}$
$\text{R.H.S}=\lim\limits_{\text{x}\rightarrow5}(4+\text{x})$
$=4+5=9\ \cdots{\text{(iii})}$
Substituting (ii) and (iii) in (i)
$9\text{a}^8=9$
$\Rightarrow\text{a}^{8}=1$
$\Rightarrow\text{a}^4=1$
$\text{a}^2=1$
$\Rightarrow\text{a} = 1\text{ and a} = -1$

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