Download App

Limits — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits3 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{1}}\frac{\text{x}^{15}-1}{\text{x}^{10}-1}$
✓

Answer

$\lim\limits_{\text{x}\rightarrow{1}}\frac{\text{x}^{15}-1}{\text{x}^{10}-1}$$=\lim\limits_{\text{x}\rightarrow{1}}\frac{\frac{\text{x}^{15}-1^{15}}{\text{x}-1}}{\frac{\text{x}^{10}-1^{10}}{\text{x}-1}}$ [Dividing numerator and denominator by x - 1]
$=\frac{\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}^{15}-1^{15}}{\text{x}-1}}{\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}^{10}-1^{10}}{\text{x}-1}}$
Applying formula $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}$ in numerator and $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{m}}-\text{a}^{\text{m}}}{\text{x}-\text{a}}=\text{ma}^{\text{m}-1}$ in denominator
$\Rightarrow\text{n}=15,\text{m}=10$
$\Rightarrow\frac{\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}^{15}-1^{15}}{\text{x}-{1}}}{\lim\limits_{\text{x}\rightarrow1}\frac{\text{x}^{10}-1^{10}}{\text{x}-1}}=\frac{15(1)^{15-1}}{10(1)^{10-1}}$
$=\frac{15}{10}$
$=\frac32$

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 "3 Marks Question" from Limits→Limits — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that:
$(\text{A}\cap\text{B})'=\text{A'}\cup\text{B'}$
→
Let A = {x : x $\in$ N}, B = {x : x = 2n, n $\in$ N}, C = {x : x = 2n - 1, n $\in$ N} and D = {x : x is a prime natural number}. Find:
$\text{B}\cap\text{D}$
→
Find $(x+1)^6+(x-1)^6$. Hence or otherwise evaluate $(\sqrt{2}+1)^6+(\sqrt{2}-1)^6$
→
Show that the following four conditions are equivalent :
  1. $A \subset B$
  2. $A – B = \phi$
  3. $A \cup B = B$
  4. $A \cap B = A$
→
Let $X = {1, 2, 3, 4}$ and $Y = {1, 5, 9, 11, 15, 16}.$ Determine which of the following sets are functions from $X$ to $Y:$
$f_3 = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}$
→
Solve the following equations:
$\text{cosec }\text{x}=1+\cot\text{x}$
→
If A.M. and G.M. of roots of a quadratic equation are $8$ and $5$ respectively then obtain the quadratic equation.
→
If a, b, c, d and p are different real numbers such that:
$(\text{a}^2+\text{b}^2+\text{c}^2)\text{p}^2-2(\text{ab}+\text{bc}+\text{cd})\text{p}+(\text{b}^2+\text{c}^2+\text{a}^2)\le0,$ then show that a, b, c and d are in G.P.
→
Find the values of the other five trigonometric functions in the following:
$\cot\text{x}=\frac{12}{5},$ x in quadrant III.
→
Find the sum of integers from $1$ to $100$ that are divisible by $2$ or $5.$
→