Evaluate the limit-limx- 1x15-1x10-1

At x→ 1, x^15 1x^10 1 becomes 00 form Now, lim_x→ 1x^15 1x^10 1 =lim_x→ 1x^15 1^15x^10 1^10 Divide and multiply by (x 1), we get =lim_x→ 1 [x^15 1^15x 1×x 1 ...

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Standard XI

Mathematics

Algebra of Limits

Evaluate the ...

Question

Evaluate the limit:
$lim x \to 1 \frac{x^{15} - 1}{x^{10} - 1}$

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Solution

At $x \to 1$ ,

$\frac{x^{15} - 1}{x^{10} - 1}$ becomes $\frac{0}{0}$ form

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

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

Divide and multiply by $(x - 1)$ , we get
$= lim x \to 1 [\frac{x^{15} - 1^{15}}{x - 1} \times \frac{x - 1}{x^{10} - 1^{10}}]$

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

$= \frac{15 (1)^{15 - 1}}{10 (1)^{10 - 1}} [∵ lim x \to 1 \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}]$

$= \frac{3}{2}$

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

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Algebra of Limits

Standard XI Mathematics

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Evaluate the limit: limx→1x15−1x10−1

At x→1, x15−1x10−1 becomes 00 form Now, limx→1x15−1x10−1 =limx→1x15−115x10−110 Divide and multiply by (x−1), we get =limx→1[x15−115x−1×x−1x10−110] =limx→1[x15−115x−1]limx→1[x10−110x−1] =15(1)15−110(1)10−1[∵limx→1xn−anx−a=nan−1] =32

Evaluate the limit:
$lim x \to 1 \frac{x^{15} - 1}{x^{10} - 1}$