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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits4 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}$
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Answer

$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}$$\Rightarrow\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}-\text{a}}}{\frac{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}{\text{x}-\text{a}}}$ [Dividing numrator and denominator by x - a]
$=\frac{\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}-\text{a}}}{\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}{\text{x}-\text{a}}}$
Applying 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, $\text{n}=\frac57$ is numerator and applying $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^\text{m}-\text{a}^\text{m}}{\text{x}-\text{a}}=\text{ma}^{\text{m}-1}$ in denominator, where $\text{m}=\frac27$
$\Rightarrow\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}-\text{a}}}{\frac{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}{\text{x}-\text{a}}}=\frac{\frac{5}{7}\text{a}^{\frac{5}{7}-1}}{\frac{2}{7}(\text{a})^{\frac{2}{7}-1}}$
$=\frac{\frac{5}{7}\text{a}^{\frac{-2}{7}}}{\frac{2}{7}\text{a}^{\frac{-5}{7}}}$
$=\frac{5}{2}\text{a}^{\frac{-2}{7}+\frac{5}{7}}$
$=\frac{5}{2}\text{a}^{\frac{3}{7}}$

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