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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits4 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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