Download App

Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits4 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\frac{\pi}{8}}\frac{\cot4\text{x}-\cos4\text{x}}{(\pi-8\text{x})^3}$
✓

Answer

$\lim\limits_{\text{x}\rightarrow\frac{\pi}{8}}\frac{\cot4\text{x}-\cos4\text{x}}{(\pi-8\text{x})^3}$
When $\text{x}\rightarrow\frac\pi8,\frac\pi8-\text{x}\rightarrow0,$ let $\frac{\pi}{8}-\text{x}=\text{y}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\cot4\big(\frac\pi8-\text{y}\big)-\cos4\big(\frac\pi8-\text{y}\big)}{(8)^3\text{y}^3}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\cot4\big(\frac\pi2-4\text{y}\big)-\cos4\big(\frac\pi2-4\text{y}\big)}{(8)^3\text{y}^3}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\tan4\text{y}-\sin4\text{y}}{(8)^2\text{y}^3}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\frac{\sin4\text{y}}{\cos4\text{y}}-\sin4\text{y}}{8^3\text{y}^3}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\sin4\text{y}-\sin4\text{y}\cos4\text{y}}{\cos4\text{y}\times\text{y}^3\times8^3}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\sin4\text{y}\big(2\sin^22\text{y}\big)}{\cos4\text{y}\times\text{y}^3\times8^3}$
$=\frac{2}{8^3}\lim\limits_{\text{y}\rightarrow{0}}\frac{\sin4\text{y}}{\text{y}}\times\frac{\sin^22\text{y}}{\text{y}^2}\times\frac{1}{\cos4\text{y}}$
$=\frac{2}{8^3}\Big(\lim\limits_{\text{y}\rightarrow{0}}\frac{\sin4\text{y}}{4\text{y}}\times4\Big)\times\Big(\lim\limits_{\text{y}\rightarrow{0}}\frac{\sin2\text{y}}{2\text{y}}\Big)^2\times4\times\frac{1}{\lim\limits_{\text{y}\rightarrow0}\cos4\text{y}}$
$=\frac{2}{8^2}(1\times4)\times(1)\times4\times\frac11$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1,\lim\limits_{\theta\rightarrow0}\cos\theta=1\Big]$
$=\frac{2\times4\times4}{8\times8\times8}$
$\frac{1}{16}$

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 "Solve the Following Question.(4 Marks)" from Limits→Limits — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\text{a}\cos2\text{x}+\text{b}\sin2\text{x}=\text{c}$ has $\alpha$ and $\beta$ as its roots, then prove that,
$\tan\alpha+\tan\beta=\frac{2\text{b}}{\text{a+c}}$
→
There are 6 positive and 8 negative numbers. Four numbers are chosen at random, without replacement, and multiplied. Find the probability that the product is a positive number.
→
Find the value of $\frac{3+\log _{10} 343}{2+\frac{1}{2} \log _{10}\left(\frac{49}{4}\right)+\frac{1}{2} \log _{10}\left(\frac{1}{25}\right)}$
→
Prove by the method of induction, for all n ∈ N.

$1^3+3^3+5^3+\ldots .$. to $n$ terms $=n^2\left(2 n^2-1\right)$

→
Find the value of

$2 x^4+5 x^3+7 x^2-x+41$, if $x=-2-\sqrt{3 i}$

→
Prove that:
$\frac{\sin3\text{A}+\sin5\text{A}+\sin7\text{A}+\sin9\text{A}}{\cos3\text{A}+\cos5\text{A}+\cos7\text{A}\cos9\text{A}}=\tan6\text{A}$
→
$\text{a}^2=(\text{b + c})^2-4\text{bc}\cos^2\frac{\text{A}}{2}$
→
In how many ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs of the same colour are indistinguishable?
→
Let f and g be two real functions defined by $\text{f(x)}=\sqrt{\text{x}+1}$ and $\text{g(x)}=\sqrt{9-\text{x}^2}$ Then describe the following functions:
$\frac{5}{\text{g}}$
→
A number is drawn at random from the numbers 1 to 50. Find the probability that it is divisible by 2 or 3 or 10.
→