Question:
$\lim _{x \rightarrow 1-} \frac{\sqrt{\pi}-\sqrt{2 \sin ^{-1} x}}{\sqrt{1-x}}$ ie equal to :
$\frac{1}{\sqrt{2 \pi}}$
$\sqrt{\frac{\pi}{2}}$
$\sqrt{\frac{2}{\pi}}$
$\sqrt{\pi}$
Correct Option: , 3
Solution:
$\lim _{x \rightarrow 1^{-}} \frac{\sqrt{\pi}-\sqrt{2 \sin ^{-1} x}}{\sqrt{1-x}} \times \frac{\sqrt{\pi}+\sqrt{2 \sin ^{-1} x}}{\sqrt{\pi}+\sqrt{2 \sin ^{-1} x}}$
$\lim _{x \rightarrow 1^{-}} \frac{2\left(\frac{\pi}{2}-\sin ^{-1} x\right)}{\sqrt{1-x} \cdot\left(\sqrt{\pi}+\sqrt{2 \sin ^{-1} x}\right)}$
$\lim _{x \rightarrow 1^{-}} \frac{2 \cos ^{-1} x}{\sqrt{1-x}} \cdot \frac{1}{2 \sqrt{\pi}}$
Put $x=\cos \theta$
$\lim _{\theta \rightarrow 0^{+}} \frac{2 \theta}{\sqrt{2} \sin \left(\frac{\theta}{2}\right)} \cdot \frac{1}{2 \sqrt{\pi}}=\sqrt{\frac{2}{\pi}}$
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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,
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