Download App

Limits and Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits and Derivatives5 Marks
Question
Evaluate:
$\lim\limits_{\text{x} \rightarrow 0}\frac{2\sin\text{x}-\sin2\text{x}}{\text{x}^{3}}$ 
✓

Answer

Given that $\lim\limits_{\text{x} \rightarrow 0}\frac{2\sin\text{x}-\sin2\text{x}}{\text{x}^{3}}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{2\sin\text{x}-2\sin\text{x}\cos\text{x}}{\text{x}^{3}}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{2\sin\text{x}-(1-\cos\text{x})}{\text{x}^{3}}$
 $=\lim\limits_{\text{x} \rightarrow 0}\frac{2\sin\text{x}}{\text{x}}\Big(\frac{1-\cos\text{x}}{\text{x}^{2}}\Big)$
$=\lim\limits_{\text{x} \rightarrow 0}2\Big(\frac{\sin\text{x}}{\text{x}}\Big)\bigg(\frac{\frac{2\sin^{2}\text{x}}{2}}{\text{x}^{2}}\bigg)$
$=\lim\limits_{\text{x} \rightarrow 0}2\Big(\frac{\sin\text{x}}{\text{x}}\Big)\Bigg(2\frac{\sin^{2}\frac{\text{x}}{2}}{\frac{\text{x}^{4}}{4}}\times\frac{1}{4}\Bigg)$
 $=\lim\limits_{\text{x} \rightarrow 0}2\Big(\frac{\sin\text{x}}{\text{x}}\Big)\Bigg(2\frac{\sin^{2}\frac{\text{x}}{2}}{\frac{\text{x}^{4}}{4}}\Bigg)^{2}.\frac{1}{4}$
$=\lim\limits_{\text{x} \rightarrow 0}\frac{4}{4}\Big(\frac{\sin\text{x}}{\text{x}}\Big)\lim\limits_{\frac{\text{x}}{2} \rightarrow 0}\Bigg(\frac{\sin\frac{\text{x}}{2}}{\frac{\text{x}}{2}}\Bigg)$
$1.1.(1)^{2}=1$
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→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Using binomial evaluate the following:
$(98)^5$
→
Prove the following statement by principle of mathematical induction:
$3^{2n} - 1$ is divisible by $8,$ for all natural numbers $n.$
→
If $\sin\text{x}=\frac{12}{13}$ and x lies in the second quadrant, find the value of $\sec\text{x}+\tan\text{x}.$
→
If $\cos\text{x}=\frac{\cos\alpha+\cos\beta}{1+\cos\alpha\cos\beta}$ prove that $\tan\frac{\text{x}}{2}=\pm\tan\frac{\alpha}{2}\tan\frac{\beta}{2}$
→
Sketch the graphs of the following trigonometric functions:
$\phi\text{(x)}=2\cos\Big(\text{x}-\frac{\pi}{6}\Big)$
→
If P(n, 4) = 12 . P(n, 2), find n.
→
Let r and n be positive integers such that 1 < r < n. Then prove the following:
${^\text{n}\text{C}}_{\text{r}}+2\ {^\text{n}\text{C}}_{\text{r}-1}+{^\text{n}\text{C}}_{\text{r}-2}={^\text{n+2}\text{C}}_{\text{r}}$
→
IQ of a person is given by the formula
$\mathrm{IQ}=\frac{\mathrm{MA}}{\mathrm{CA}} \times 100$
where MA is mental age and CA is chronological age. If 80 $\le$ IQ $\le$ 140 for a group of 12 years old children, find the range of their mental age.
→
Find the equation of the circle which passes through the points $(2, 3)$ and $(4,5)$ and the centre lies on the straight line $y - 4x + 3 = 0$
→
Prove that:
$\frac{\sin11\text{A}\sin\text{A}+\sin7\text{A}\sin3\text{A}}{\cos11\text{A}\sin\text{A}+\cos7\text{A}\sin3\text{A}}=\tan8\text{A}$
→