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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits4 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{5}-\text{a}^5}{\text{x}-\text{a}}=405,$ find all possible value of a.
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Answer

If $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{5}-\text{a}^5}{\text{x}-\text{a}}=405\ \cdots{\text{(i)}}$
$\text{L.H.S}=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{5}-\text{a}^5}{\text{x}-\text{a}}$
$=5(\text{a})^{5-1}$
$=5\text{a}^{4}$
It is given that $5\text{a}^4=405$
$\Rightarrow5\text{a}^4=405$
$\text{a}^4=\frac{405}{5}=81$
$\text{a}^4=(3)^4,\text{a}^2=9$
$\text{a}=\pm3$
$\Rightarrow\text{a}=3 \text{ and }\text{a}=-3$

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