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NCERT Class XI Mathematics - Limits and Derivatives - Solutions

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Question : 10 of 72
Marks: +1, -0
lim⁡z→1z1/3−1z1/6−1\lim\limits_{z\rightarrow 1}\frac{z^{1/3}-1}{z^{1/6}-1}
Solution:  
We have , lim⁡z→1z1/3−1z1/6−1\lim\limits_{z\rightarrow 1}\frac{z^{1/3}-1}{z^{1/6}-1} (0/0 form)
= lim⁡z→1(z1/6)2−1z1/6−1\lim\limits_{z\rightarrow 1}\frac{(z^{1/6})^2-1}{z^{1/6}-1} = lim⁡z→1(z1/6−1)(z1/6+1)z1/6−1\lim\limits_{z\rightarrow 1}\frac{(z^{1/6}-1)(z^{1/6}+1)}{z^{1/6}-1}
= lim⁡z→1(z1/6+1)\lim\limits_{z\rightarrow 1}(z^{1/6}+1) = 11/61^{1/6} + 1 = 1 + 1 = 2.
Alternative solution :
lim⁡z→1z1/3−1z1/6−1\lim\limits_{z\rightarrow 1}\frac{z^{1/3}-1}{z^{1/6}-1} = lim⁡z→1z1/3−1z−1×z−1z1/6−1\lim\limits_{z\rightarrow 1}\frac{z^{1/3}-1}{z-1} \times \frac{z-1}{z^{1/6}-1}
= 13(1)(1/3)−1\frac{1}{3} (1)^{(1/3)-1} × 1(1/6)(1)(1/6)−1\frac{1}{(1/6)(1)^{(1/6)-1}}
= 13×61\frac{1}{3} \times \frac{6}{1} = 2
Using lim⁡x→axn−anx−a\lim\limits_{x\rightarrow a}\frac{x^n-a^n}{x-a} = nan−1na^{n-1}
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