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Limits and Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits and Derivatives3 Marks
Question
Evaluate:
$\lim\limits_{\text{x} \rightarrow 1}\frac{\text{x}^{7}-2\text{x}^{5}+1}{\text{x}^{3}-3\text{x}^{2}+2}$
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Answer

Given that $\lim\limits_{\text{x} \rightarrow 1}\frac{\text{x}^{7}-2\text{x}^{5}+1}{\text{x}^{3}-3\text{x}^{2}+2}$
$=\lim\limits_{\text{x} \rightarrow 1}\frac{\text{x}^{7}-\text{x}^{5}-\text{x}^{5}+1}{\text{x}^{3}-\text{x}^{2}-2\text{x}^{2}+2}$
$=\lim\limits_{\text{x} \rightarrow 1}\frac{\text{x}^{5}(\text{x}^{2}-1)-1(\text{x}^{5}-1)}{\text{x}^{2}(\text{x}-1)-2(\text{x}^{2}-1)}$
Dividing the numerator and denominator by (x - 1) we get
 $=\lim\limits_{\text{x} \rightarrow 1}\frac{\text{x}^{5}\Big(\frac{\text{x}^{2}-1}{\text{x}-1}\Big)-1\Big(\frac{\text{x}^{5}-1}{\text{x}-1}\Big)}{\text{x}^{2}\Big(\frac{\text{x}-1}{\text{x}-1}\Big)-2\Big(\frac{\text{x}^{2}-(1)^{5}}{\text{x}-1}\Big)}$
$=\frac{\lim\limits_{\text{x} \rightarrow 1}\text{x}^{5}(\text{x}+1)-\lim\limits_{\text{x} \rightarrow 1}\Big(\frac{\text{x}^{5}-(1)^{5}}{\text{x}-1}\Big)}{\lim\limits_{\text{x} \rightarrow 1}\text{x}^{2}-2\lim\limits_{\text{x} \rightarrow 1}(\text{x}+1)}$
$=\frac{1(2)-5.(1)^{5-1}}{1-2(2)}$
$=\frac{2-5}{1-4}=\frac{-3}{-3}=1$
Hence, the required answer is 1.

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