Download App

Limits — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits5 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\cos\text{ax}-\cos\text{bx}}{\cos\text{cx}-\cos\text{dx}}$
✓

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\cos\text{ax}-\cos\text{bx}}{\cos\text{cx}-\cos\text{dx}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{{\Big(-2\sin\big(\frac{\text{a}+\text{b}}{2}\big)\times\sin\big(\frac{\text{a}+\text{b}}{2}\big)\text{x}\Big)}}{-2\sin\big(\frac{\text{c}+\text{d}}{2}\big)\times\big(\frac{\text{c}-\text{d}}{2}\big)\text{x}}$
$=\frac{\lim\limits_{\text{x}\rightarrow0}{\sin\big(\frac{\text{a}+\text{b}}{2}\big)\text{x}\sin\big(\frac{\text{a}+\text{b}}{2}\big)\text{x}}}{\lim\limits_{\text{x}\rightarrow0}{\sin\big(\frac{\text{c}+\text{d}}{2}\big)\text{x}\sin\big(\frac{\text{c}+\text{d}}{2}\big)\text{x}}}$
$=\frac{\Bigg(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\big(\frac{\text{a}+\text{b}}{2}\big)\text{x}}{\big(\frac{\text{a}+\text{b}}{2}\big)\text{x}}\times\big(\frac{\text{a}+\text{b}}{2}\big)\text{x}\Bigg)\Bigg(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\big(\frac{\text{a}-\text{b}}{2}\big)\text{x}}{\big(\frac{\text{a}-\text{b}}{2}\big)\text{x}}\times\big(\frac{\text{a}-\text{b}}{2}\big)\text{x}\Bigg)}{\Bigg(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\big(\frac{\text{c}+\text{d}}{2}\big)\text{x}}{\big(\frac{\text{c}+\text{d}}{2}\big)\text{x}}\times\big(\frac{\text{c}+\text{d}}{2}\big)\text{x}\Bigg)\Bigg(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\big(\frac{\text{c}-\text{d}}{2}\big)\text{x}}{\big(\frac{\text{c}-\text{d}}{2}\big)\text{x}}\times\big(\frac{\text{c}-\text{d}}{2}\big)\text{x}\Bigg)}$
$=\frac{(\text{a}+\text{b})(\text{a}-\text{b})}{(\text{c}+\text{d})(\text{c}-\text{d})}$ $\Big[\because\ \lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1\Big]$
$=\frac{\text{a}^2-\text{b}^2}{\text{c}^2-\text{d}^2}$

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 "5 Marks Questions" 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

Solve the equation$ |z| = z + 1 + 2i.$
→
$1$f the sum of p terms of an $AP$. is $q$ and the sum of $q$ terms is $p$, then show that the sum of $p +q$ terms is $-(p + q)$. Also$,$ find the sum of first $p - q$ terms $($where$, p > q).$
→
Find the middle terms(s) in the expansion of:
$\Big(2\text{x}-\frac{\text{x}^{2}}{4}\Big)^{9}$
→
Find the angles between the following pairs of straight lines:
3x + 4y - 7 = 0 and 4x - 3y + 5 = 0
→
If $0\leq\text{x}\leq\pi$ and x lies in the IInd quadrant such that $\sin\text{x}=\frac{1}{4}.$ Find the values of $\cos\frac{\text{x}}{2},\sin\frac{\text{x}}{2}$ and $\tan\frac{\text{x}}{2}$
→
Show that the point $(\text{x},\ \text{y})$ given by $\text{x}=\frac{2\text{at}}{1+\text{t}^2}$ and $\text{y}=\text{a}\Big(\frac{1-\text{t}^2}{1+\text{t}^2}\Big)$2 lies on a circle for all real values of t such that $-1\leq\text{t}\leq1,$ where a is any given real number.
→
Find the equation of an ellipse whose axes lie along coordinate axes, which passes through the point (-3, 1) and has eccentricity equal to $\sqrt{\frac{2}{5}}.$
→
For all sets A, B and C, show that $(\text{A} - \text{B}) \cap (\text{C} - \text{B}) = \text{A} - (\text{B} \cup \text{C})$
Determine whether each of the statement in Exercises 13 - 17 is true or false. Justify your answer.
→
Find the equation of the line which passes through the point (-4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5 : 3 by this point.
→
Sketch the graphs of the following functions:
$\text{f(x)}=2\sec\pi\text{x}$
→