Lim_x → 1x^15 1/x^10 1 =lim_x → 1x^15 1/x 1/x^10 1^10/x 1 [Dividing numerator and denominator by (x 1)] =lim_x → 1x^15 1^15/x 1/lim_x → 1x^10 1^10/x 1 Applying ...

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Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → 1x15-...

Question

${lim}_{x \to 1} \frac{x^{15} - 1}{x^{10} - 1}$

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Solution

${lim}_{x \to 1} \frac{x^{15} - 1}{x^{10} - 1}$

$= {lim}_{x \to 1} \frac{\frac{x^{15} - 1}{x - 1}}{\frac{x^{10} - 1^{10}}{x - 1}}$ [Dividing numerator and denominator by (x-1)]

$= \frac{{lim}_{x \to 1} \frac{x^{15} - 1^{15}}{x - 1}}{{lim}_{x \to 1} \frac{x^{10} - 1^{10}}{x - 1}}$

Applying the formula ${lim}_{x \to a} \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}$ in numerator and denominator

$\Rightarrow \frac{{lim}_{x \to 1} \frac{x^{15} - 1^{15}}{x - 1}}{{lim}_{x \to 1} \frac{x^{10} - 1^{10}}{x - 1}} = \frac{15 (1)^{15 - 1}}{10 (1)^{10 - 1}}$

$= \frac{15}{10} = \frac{3}{2}$
Hence, answer is $\frac{3}{2}$

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limx→1x15−1x10−1

${lim}_{x \to 1} \frac{x^{15} - 1}{x^{10} - 1}$