Question

Calculate the limit.

$\displaystyle \lim_{n \to \infty} e^{-\sqrt{n}\ x}\left(1-\dfrac{x}{\sqrt{n}}\right)^{-n}$

Answer 1

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Answer 2

$\huge\bf\red{SoluTion}$

(a) The equation of progressive wave travelling from right to left is given by the displacement function:

y(x,t)=asin(ωt+kx+ϕ) ... (i)

The given equation is:

y(x,t)=3.0sin(36t+0.018x+

4π ) ...(ii)

On comparing both the equations, we find that equation (ii) represents a travelling wave, propgating from right to left.

Now using equations (i) and (ii), we can write:

ω=36 rad/s and k= 0.018 m

−1

We know that:

v=ω/2π and λ=2π/k

Also,

v=fλ

∴v=(ω/2π)×(2π/k)=ω/k

=36/0.018=2000cm/s=20m/s

Hence, the speed of the given travelling wave is 20 m/s.

(b) Amplitude of the given wave, a=3cm

Frequency of the given wave:

f=ω/2π=36/2×3.14=573Hz

(c) On comparing equations (i) and (ii), we find that the intial phase angle, ϕ=π/4

(d) The distance between two successive crests (or troughs) is equal to the wavelength of the wave.

Wavelength is given by the relation: k=2π/λ

∴λ=2π/k=2×3.14/0.018=348.89cm=3.49m