$\lim _{x \rightarrow 2} \frac{x^{2}-4}{\sqrt{3 x-2}-\sqrt{x+2}}$
$\lim _{x \rightarrow 2} \frac{x^{2}-4}{\sqrt{3 x-2}-\sqrt{x+2}}$
Now to rationalize the denominator by multiplying the given equation by its rationalizing factor we get
$\lim _{x \rightarrow 2} \frac{x^{2}-4}{\sqrt{3 x-2}-\sqrt{x+2}}=\lim _{x \rightarrow 2} \frac{x^{2}-4}{\sqrt{3 x-2}-\sqrt{x+2}} \times\left(\frac{\sqrt{3 x-2}+\sqrt{x+2}}{\sqrt{3 x-2}+\sqrt{x+2}}\right)$
Taking $(x-2)(x+2)$ as common we get
$\lim _{x \rightarrow 2} \frac{\left(x^{2}-4\right)(\sqrt{3 x-2}+\sqrt{x+2})}{(3 x-2)-(x+2)}=\lim _{x \rightarrow 2} \frac{(x-2)(x+2)(\sqrt{3 x-2}+\sqrt{x+2})}{2 x-4}$
$\lim _{x \rightarrow 2} \frac{x^{2}-4}{\sqrt{3 x-2}-\sqrt{x+2}}=\lim _{x \rightarrow 2} \frac{x^{2}-4}{\sqrt{3 x-2}-\sqrt{x+2}} \times\left(\frac{\sqrt{3 x-2}+\sqrt{x+2}}{\sqrt{3 x-2}+\sqrt{x+2}}\right)$
Taking $(x-2)(x+2)$ as common we get
$\lim _{x \rightarrow 2} \frac{\left(x^{2}-4\right)(\sqrt{3 x-2}+\sqrt{x+2})}{(3 x-2)-(x+2)}=\lim _{x \rightarrow 2} \frac{(x-2)(x+2)(\sqrt{3 x-2}+\sqrt{x+2})}{2 x-4}$
Taking common and simplifying we get
$\lim _{x \rightarrow 2} \frac{(x-2)(x+2)(\sqrt{3 x-2}+\sqrt{x+2})}{2 x-4}=\lim _{x \rightarrow 2} \frac{(x-2)(x+2)(\sqrt{3 x-2}+\sqrt{x+2})}{2(x-2)}$
Now by applying the limit we get
$\Rightarrow \lim _{x \rightarrow 2} \frac{(x+2)(\sqrt{3 x-2}+\sqrt{x+2})}{2}=8$
$\lim _{x \rightarrow 2} \frac{x^{2}-4}{\sqrt{3 x-2}-\sqrt{x+2}}=8$
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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,