Download App

Limits and Derivatives — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSLimits and Derivatives5 Marks
Question
Evaluate:
$\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}+3\text{x}}{2\text{x}+\tan3\text{x}}$
✓

Answer

Given that $\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}+3\text{x}}{2\text{x}+\tan3\text{x}}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{\big(\frac{\sin2\text{x}+3\text{x}}{2\text{x}}\big)\times2\text{x}}{\big(\frac{2\text{x}+\tan3\text{x}}{3\text{x}}\big)\times3\text{x}}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{\Big(\frac{\sin2\text{x}}{2\text{x}}+\frac{3\text{x}}{2\text{x}}\Big)\times2\text{x}}{\Big(\frac{2\text{x}\tan3\text{x}}{3\text{x}}\Big)\times3\text{x}}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{\Big(\frac{\sin2\text{x}}{2\text{x}}+\frac{3\text{x}}{2\text{x}}\Big)\times2\text{x}}{\Big(\frac{2\text{x}}{3\text{x}}+\frac{\tan3\text{x}}{3\text{x}}\Big)\times3\text{x}}$
$=\frac{\Big(\lim\limits_{2\text{x} \rightarrow 0}\frac{\sin2\text{x}}{2\text{x}}+\frac{3}{2}\Big)}{\Big(\frac{2}{3}+\lim\limits_{3\text{x} \rightarrow 0}\frac{\tan3\text{x}}{3\text{x}}\Big)}\times\frac{2}{3}$
$=\Big(\frac{1+\frac{3}{2}}{\frac{2}{3}+1}\Big)\times\frac{2}{3}$
$=\frac{\frac{5}{2}}{\frac{5}{3}}\times\frac{2}{3}=\frac{3}{2}\times\frac{2}{3}=1$
Hence, the required answer is 1.

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 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 centre, the lengths of the axes, eccentricity, foci of the following ellipse:

$4\text{x}^2+16\text{y}^2-24\text{x}-32\text{y}-12=0$

→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{{\pi}}}\frac{1+\cos\text{x}}{\tan^2\text{ x}}$
→
If $\cos\text{x}-\sin\text{x}=\text{a}^3, \sec\text{x}-\cos\text{x}=\text{b}^3,$ than proved that a2b2 (a2 + b2) = 1.
→
The equation of the base of an equilateral triangle is x + y = 2 and its vertex is (2, -1). Find the length and equations of its sides.
→

Find the middle terms(s) in the expansion of:

$\Big(\frac{\text{x}}{\text{a}}-\frac{\text{a}}{\text{x}}\Big)^{10}$

→
Find the equations to the straight lines passing through the point (2, 3) and inclined at and angle of 45° to the line 3x + y - 5 = 0.
→
Find the equation of a line which is perpendicular to the line $\sqrt{3}\text{x}-\text{y}+5=0$ and which cuts off an intercept of 4 units with the negative direction of y-axis.
→
$\Bigg|\sin\text{x}\sin\Big(\frac{\pi}{3}-\text{x}\Big)\sin\Big(\frac{\pi}{3}+\text{x}\Big)\Bigg|\not<\frac{1}{4}$ for all values of x.
→
If $\sin\alpha+\sin\beta=\text{a}$ and $\cos\alpha+\cos\beta=\text{b},$ show that
$\cos(\alpha+\beta)=\frac{\text{b}^2-\text{a}^2}{\text{b}^2+\text{a}^2}$
→
Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:
$5\text{x}^2+4\text{y}^2=1$
→