Question.
$\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2} \frac{\pi}{3}-x}$
$\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2} \frac{\pi}{3}-x}$
solution:
Given $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}$
Now by using the formula
$\cos 6 x=1-2 \sin ^{2} 3 x$
Then the above equation becomes,
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1-\left(1-2 \sin ^{2} 3 x\right)}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}$
Again using $\sin 3 x$ formula the above equation can be written as
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sqrt{1-\left(1-2 \sin ^{2} 3 x\right)}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}=\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sqrt{2}|\sin 3 x|}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}$
Now we have
$\sin 3 x=\sin (\pi-3 x)=\sin 3\left(\frac{\pi}{3}-x\right)$
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{2}|\sin 3 x|}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin 3\left(\frac{\pi}{3}-x\right)}{\left(\frac{\pi}{3}-x\right)}$
On simplifying we get
$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin 3\left(\frac{\pi}{3}-x\right)}{\left(\frac{\pi}{2}-x\right)}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{3 \sin 3\left(\frac{\pi}{3}-x\right)}{3\left(\frac{\pi}{2}-x\right)}=3 \lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin (\pi-3 x)}{(\pi-3 x)}$
Now as $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
By substituting the limit we get
$\Rightarrow$$3 \lim _{x \rightarrow \frac{t}{2}} \frac{\sin (\pi-3 x)}{(\pi-3 x)}=3.1=3$
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}=3$
Given $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}$
Now by using the formula
$\cos 6 x=1-2 \sin ^{2} 3 x$
Then the above equation becomes,
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1-\left(1-2 \sin ^{2} 3 x\right)}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}$
Again using $\sin 3 x$ formula the above equation can be written as
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sqrt{1-\left(1-2 \sin ^{2} 3 x\right)}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}=\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sqrt{2}|\sin 3 x|}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}$
Now we have
$\sin 3 x=\sin (\pi-3 x)=\sin 3\left(\frac{\pi}{3}-x\right)$
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{2}|\sin 3 x|}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin 3\left(\frac{\pi}{3}-x\right)}{\left(\frac{\pi}{3}-x\right)}$
On simplifying we get
$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin 3\left(\frac{\pi}{3}-x\right)}{\left(\frac{\pi}{2}-x\right)}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{3 \sin 3\left(\frac{\pi}{3}-x\right)}{3\left(\frac{\pi}{2}-x\right)}=3 \lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin (\pi-3 x)}{(\pi-3 x)}$
Now as $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
By substituting the limit we get
$\Rightarrow$$3 \lim _{x \rightarrow \frac{t}{2}} \frac{\sin (\pi-3 x)}{(\pi-3 x)}=3.1=3$
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}=3$
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
Free Study Material