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

Question

Evaluate $\lim _{x \rightarrow 1} \frac{x^{15}-1}{x^{10}-1}$.

✓

Answer

$\begin{array}{l} \lim _{x \rightarrow 1} \frac{x^{15}-1}{x^{10}-1} \\ =\lim _{x \rightarrow 1} \frac{\left(x^5\right)^3-1^3}{\left(x^5\right)^2-1^2}\end{array}$
$\begin{array}{l}\quad \because a^3-b^3=(a-b)\left(a^2+b^2+a b\right) \\ =\lim _{x \rightarrow 1} \frac{\left(x^5-1\right)\left(x^{10}+x^5+1\right)}{\left(x^5-1\right)\left(x^5+1\right)} \quad[x \neq 1] \\ =\lim _{x \rightarrow 1} \frac{x^{10}+x^5+1}{x^5+1}\end{array}$
$\begin{array}{l}=\frac{1^{10}+1^5+1}{1^5+1} \\ =\frac{3}{2} .\end{array}$

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 "2 Marks Questions" from Differentiation of Derivatives→Differentiation of Derivatives — all question types→Applied Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Find decimal equivalent of the octal number $(645)_8$.

If $\log (0.57)=\overline{1} .756$, then find the value of $\log 57+\log (0.57)^3+\log \sqrt{0.57}$

Arrange each of the following words in meaningful logical order.
(i) 1. Probation 2. Interview 3. Selection 4. Appointment 5. Advertisement 6. Application
(ii) 1. Gold 2. Iron 3. Sand 4. Platinum 5. Diamond

If $\mathrm{y}=\log \{\sqrt{x-1}-\sqrt{x+1}\}$, show that $\frac{d y}{d x}=\frac{-1}{2 \sqrt{x^{2}-1}}$.

Prove that $\mathrm{A}-\mathrm{B}=\mathrm{A} \cap \mathrm{B}^{\prime}$.

Prove by the direct method for any integer ' $n$ ', $n^3-n$ is always even.

By giving the counter example, show that the following statement is false.
"If $n$ is an odd integer, then $n$ is prime"

Which of the following sentences are statements ?
Give reasons.
(i) Mathematics is fun.
(ii) The number $n$ is a positive integer

Write the converse of the statement:
If $3 \times 7=21$, then $3+7=10$

The collection of difficult topics in Mathematics is a set or not. Justify your answer.