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Limits and Derivatives — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSLimits and Derivatives3 Marks
Question
Evaluate:
$\lim\limits_{\text{x} \rightarrow \text{1}}\frac{\text{x}^{4}-\sqrt{\text{x}}}{\sqrt{\text{x}}-1}$
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Answer

Given that $\lim\limits_{\text{x} \rightarrow \text{1}}\frac{\text{x}^{4}-\sqrt{\text{x}}}{\sqrt{\text{x}}-1}$
$=\lim\limits_{\text{x} \rightarrow 1}\frac{\sqrt{\text{x}}\Big[ (\text{x})^\frac{7}{2}-1\Big]}{\sqrt{\text{x}-1}}$
$=\lim\limits_{\text{x} \rightarrow 1}\frac{\sqrt{\text{x}}\frac{\bigg[\text{x}^\frac{7}{2}-(1)^{\frac{7}{2}}\bigg]}{\text{x}-1}}{\frac{(\text{x})^\frac{1}{2}-(1)^\frac{1}{2}}{\text{x}-1}}$
Dividing the numerator and denominator of x - 1
$=\lim\limits_{\text{x} \rightarrow1}\frac{\sqrt{\text{x}}\frac{\bigg[\text{x}^\frac{7}{2}-(1)^{\frac{7}{2}}\bigg]}{\text{x}-1}}{\frac{(\text{x})^\frac{1}{2}-(1)^\frac{1}{2}}{\text{x}-1}}\times\lim\limits_{\text{x} \rightarrow 1}\sqrt{\text{x}}$
$=\frac{\frac{7}{2}(1)^{\frac{7}{2}-1}}{\frac{1}{2}(1)^{\frac{1}{2}-1}}\times\sqrt{1}$
$=\frac{\frac{7}{2}}{\frac{1}{2}}=7$
Hence, the required answer is 7.

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