Evaluate: _ x 1 x^ 15 - 1 x^ 10 - 1 - Vidyadip

Question

Evaluate: $\mathop {\lim }\limits_{x \to 1} \frac{{{x^{15}} - 1}}{{{x^{10}} - 1}}$

✓

Answer

We have,
$\mathop {\lim }\limits_{x \to 1} \frac{{{x^{15}} - 1}}{{{x^{10}} - 1}}$ = $\mathop {\lim }\limits_{x \to 1} \frac{{{x^{15}} - 1}}{{{x^{10}} - 1}} \times \frac{{(x - 1)}}{{(x - 1)}}$
= $\mathop {\lim }\limits_{x \to 1} \frac{{{x^{15}} - 1}}{{x - 1}} \div \mathop {\lim }\limits_{x \to 1} \frac{{{x^{10}} - 1}}{{x - 1}}$
$\mathop {\lim }\limits_{x \to 1} \frac{{{x^{15}} - {{(1)}^{15}}}}{{x - 1}} \div \mathop {\lim }\limits_{x \to 1} \frac{{{x^{10}} - {{(1)}^{10}}}}{{x - 1}}$
= 15(1)¹⁴ $\div$ 10(1)⁹ [$\because {\mathop {\lim }\limits_{x \to a} \frac{{{x^n} - {a^n}}}{{x - a}}}$ = na^{n - 1}]
= $\frac { 15 } { 10 } = \frac { 3 } { 2 }$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "1 Marks Question" from Limits and Derivatives→Limits and Derivatives — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the component statements of the following compound statements:
The sky is blue and the grass is green.

Find the degree measure to the radian measures $\left( {use\;\pi \frac{{22}}{7}} \right)\frac{{7\pi }}{6}$

Find the union pair of set: A = {x : x is a natural number and 1 < x $\leq$ 6} and B = {x : x is a natural number and 6 < x < 10}

Is the set of lines which are parallel to the x-axis finite or infinite?

Which of the following cannot be valid assignment of probability for elements or outcomes of sample space s = W₁, W₂,W_3, W₄, W₅, W₆, W₇.

Elementary events:	W₁	W₂	W₃	W₄	W₅	W₆	W₇
	0.7	0.6	0.5	0.4	0.3	0.2	0.1

Calculate the probability of one Sunday in a week.

Write the component statements of the following compound statements and check whether the compound statement is true or false.
57 is divisible by 2 or 3.

A coin is tossed and then a die is thrown Describe the sample space for this experiment.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{{\text{x}+\text{a}}}$

Write true/false for the following sets?
(i) $\phi \in \phi$
(ii) $0 \in \phi$
(iii) $\phi=\{0\}$