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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits4 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}^{\text{n}}-3^\text{n}}{\text{x}-\text{3}}=108,$ find the value of n.
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Answer

$\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}^{\text{n}}-3^\text{n}}{\text{x}-\text{3}}=108$$\text{L.H.S}=\lim\limits_{\text{x}\rightarrow{3}}\frac{\text{x}^{\text{n}}-3^{\text{n}}}{\text{x}-3}$
Applying the formula $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}$
Here, n = n, a = 3
$\Rightarrow{\lim\limits_{\text{x}\rightarrow{3}}}\frac{\text{x}^{\text{n}}-3^\text{n}}{\text{x}-3}=\text{n}(3)^{\text{n}-1}$
It is given that $\text{n}(3)^{\text{n}-1}=108$
$\Rightarrow\text{n}(3)^{\text{n}-1}=2\times2\times3\times3\times3$
$=(2)^2\times(3)^3$
$=4(3)^{4-1}$
$\Rightarrow\text{n}=4$

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