$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 x}{x-\frac{\pi}{2}}$
$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 x}{x-\frac{\pi}{2}}$
At $x=\frac{\pi}{2}$, the value of the given function takes the form $\frac{0}{0}$
Now, put $x-\frac{\pi}{2}=y$ so that $x \rightarrow \frac{\pi}{2}, y \rightarrow 0$
$\therefore \lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 x}{x-\frac{\pi}{2}}=\lim _{y \rightarrow 0} \frac{\tan 2\left(y+\frac{\pi}{2}\right)}{y}$
$=\lim _{y \rightarrow 0} \frac{\tan (\pi+2 y)}{y}$
$=\lim _{y \rightarrow 0} \frac{\tan 2 y}{y} \quad[\tan (\pi+2 y)=\tan 2 y]$
$=\lim _{y \rightarrow 0} \frac{\sin 2 y}{y \cos 2 y}$
$=\lim _{y \rightarrow 0}\left(\frac{\sin 2 y}{2 y} \times \frac{2}{\cos 2 y}\right)$
$=\left(\lim _{2 y \rightarrow 0} \frac{\sin 2 y}{2 y}\right) \times \lim _{y \rightarrow 0}\left(\frac{2}{\cos 2 y}\right) \quad[y \rightarrow 0 \Rightarrow 2 y \rightarrow 0]$
$=1 \times \frac{2}{\cos 0} \quad\left[\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\right]$
$=1 \times \frac{2}{1}$
$=2$
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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,