Question.
$\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}$
$\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}$
solution:
Given $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}$
We know that
$\cot ^{2} x=\operatorname{cosec}^{2} x-1$
By using this in given equation we get
$\Rightarrow$ $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\left(\operatorname{cosec}^{2} x-1\right)-3}{\operatorname{cosec} x-2}=\lim _{x \rightarrow \frac{\pi}{4}} \frac{\operatorname{cosec}^{2} x-4}{\operatorname{cosec} x-2}$
Again using $a^{2}-b^{2}$ identity the above equation can be written as
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{6}} \frac{\operatorname{cosec}^{2} x-4}{\operatorname{cosec} x-2}=\lim _{x \rightarrow \frac{\pi}{6}} \frac{(\operatorname{cosec} x-2)(\operatorname{cosec} x+2)}{\operatorname{cosec} x-2}$
On simplification and applying the limits we get
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{6}} \frac{(\operatorname{cosec} x-2)(\operatorname{cosecx}+2)}{\operatorname{cosec} x-2}=\lim _{x \rightarrow \frac{\pi}{6}}(\operatorname{cosecx}+2)=2+2=4$
$\Rightarrow$$\lim _{x \rightarrow \frac{-1}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}=4$
Given $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}$
We know that
$\cot ^{2} x=\operatorname{cosec}^{2} x-1$
By using this in given equation we get
$\Rightarrow$ $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\left(\operatorname{cosec}^{2} x-1\right)-3}{\operatorname{cosec} x-2}=\lim _{x \rightarrow \frac{\pi}{4}} \frac{\operatorname{cosec}^{2} x-4}{\operatorname{cosec} x-2}$
Again using $a^{2}-b^{2}$ identity the above equation can be written as
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{6}} \frac{\operatorname{cosec}^{2} x-4}{\operatorname{cosec} x-2}=\lim _{x \rightarrow \frac{\pi}{6}} \frac{(\operatorname{cosec} x-2)(\operatorname{cosec} x+2)}{\operatorname{cosec} x-2}$
On simplification and applying the limits we get
$\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{6}} \frac{(\operatorname{cosec} x-2)(\operatorname{cosecx}+2)}{\operatorname{cosec} x-2}=\lim _{x \rightarrow \frac{\pi}{6}}(\operatorname{cosecx}+2)=2+2=4$
$\Rightarrow$$\lim _{x \rightarrow \frac{-1}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}=4$
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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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