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Limits — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits3 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{(\text{x}+2)^{\frac{5}{2}}-(\text{a}+2)^{\frac{5}{2}}}{\text{x}-\text{a}}$
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Answer

$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{(\text{x}+2)^{\frac{5}{2}}-(\text{a}+2)^{\frac{5}{2}}}{\text{x}-\text{a}}$$=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{(\text{x}+2)^{\frac{5}{2}}-(\text{a}+2)^{\frac{5}{2}}}{(\text{x}+2)-(\text{a}+2)}$
$=\lim\limits_{\text{y}\rightarrow{\text{b}}}\frac{\text{y}^\frac{5}{2}-\text{b}^\frac{5}{2}}{\text{y}-\text{b}},$ where x + 2 = y and a + 2 = b
$=\frac{5}{2}\text{b}^{\frac{5}{2}-1}$ $\Big[\text{Using formula}\ \lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}\Big]$
$=\frac{5}{2}(\text{a}+2)^{\frac{5}{2}-1}$
$=\frac{5}{2}(\text{a}+2)^{\frac{3}{2}}$

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