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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits4 Marks
Question
Evaluate the following limit: If $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{3}-\text{a}^3}{\text{x}-\text{a}}=\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}^4-1}{\text{x}-1},$ find all possible value of a.
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Answer

If $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{3}-\text{a}^3}{\text{x}-\text{a}}=\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}^4-1}{\text{x}-1}\ \cdots{\text{(i})}$ $\text{L.H.S}=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{3}-\text{a}^3}{\text{x}-\text{a}}$ $=3(\text{a})^{3-1}$ $=3\text{a}^{2}\ \cdots{\text{(ii})}$ $\text{R.H.S}=\lim\limits_{\text{x}\rightarrow5}\frac{\text{x}^4-1}{\text{x}-1}$ $=\lim\limits_{\text{x}\rightarrow5}\frac{\text{x}^4-1}{\text{x}-1}$ $=4(1)^{4-1}$ $=4\ \cdots{(\text{iii})}$ Substituting (ii) and (iii) in (i), $3\text{a}^8=4$ $\Rightarrow\text{a}^{2}=\frac43$ $\Rightarrow\text{a}=\pm\frac{2}{\sqrt{3}}$

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