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Limits — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits5 Marks
Question
Evaluate the following limit:
If $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{3}-\text{a}^3}{\text{x}-\text{a}}=\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}^4-1}{\text{x}-1},$ 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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