Download App

Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits4 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}(\cos3\text{x}-\cos\text{x})}{\text{x}^3}$
✓

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}(\cos3\text{x}-\cos\text{x})}{\text{x}^3}$ $=\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}\Big(-2\sin\big(\frac{3\text{x}+\text{x}}{2}\big)\sin\big(\frac{3\text{x}-\text{x}}{2}\big)\Big)}{\text{x}^3}$ $=\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}(-2\sin2\text{x}\sin\text{x})}{\text{x}^3}$ $=\frac{-2\lim\limits_{\text{x} \rightarrow0}\sin2\text{x}\times\lim\limits_{\text{x} \rightarrow0}\sin2\text{x}\times\lim\limits_{\text{x} \rightarrow0}\sin\text{x}}{\text{x}^3}$ $=-2\Big(\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}}{2\text{x}}\times2\Big)\times\Big(2\lim\limits_{\text{x} \rightarrow0}\frac{\sin2\text{x}}{2\text{x}}\Big)\times\Big(\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}}{\text{x}}\Big)$ $=-2(1\times2)\times(2)\times(1)$ $=-8$

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 "(Each question 4 marks)" from Limits→Limits — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{(1+\text{x})^{{6}}-1}{(1+\text{x})^2-1}$
→
Differentiate the following from first principle: $\frac{1}{\sqrt{3-\text{x}}}$
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow-2}\frac{\text{x}^3+\text{x}^2+4\text{x}+12}{\text{x}^3+3\text{x}+2}$
→
Reduce each of the following expressions to the sine and cosin of a single expression: $\cos\text{x}-\sin\text{x}$
→
Find the equations of the hyperbola satisfying the given conditions. Foci $(\pm3\sqrt{5},0),$ the latus rectum is of length $8$.
→
Sketch the graphs of the following functions: $\text{f(x)}=\sec^2\text{x}$
→
In each of the followin find the equation of the hyperbola satisying the given conditions: vertices $(0, \pm5), $ foci $(0,\pm8)$ [NCERT]
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{2}{3}}-\text{a}^{\frac{2}{3}}}{\text{x}^{\frac{3}{4}}-\text{a}^{\frac{3}{4}}}$
→
If $\text{T}_\text{n}=\sin^\text{n}\text{x}+\cos^\text{n}\text{x},$ Prove that $6\text{T}_{10}-15\text{T}_8+10\text{T}_6-1=0$
→
20 cards are numbered form 1 to 20. card is drawn at random. what is the probability that trhe number on the card is:
  1. A multiple of 4?
  2. Not a multiple of 4?
  3. odd?
  4. Greather than 12?
  5. Divisible by 5?
  6. Not a multiple of 6?
→