Question

A knuckleball thrown with a different grip than that of the problem above has left/right

position as it crosses the plate given by

f(W) =

(0.625/w2)(1-sin (2.72w+π/2))

Again, use graphical and tabular evidence to make a conjecture about the lim f(w) as w approaches 0 from the right.

Answer 1

Solution:

Graph shows that when $\omega\rightarrow0^+, f(w)=2.312$

$f(w)=\dfrac{0.625}{\omega^2}[1-\sin(2.72\omega+\dfrac {\pi}2)] \\\Rightarrow f(w)=\dfrac{0.625}{\omega^2}[1-\cos(2.72\omega)] \\\Rightarrow \lim_{ \omega\rightarrow0^+}f(w)=\lim_{ \omega\rightarrow0^+}\dfrac{0.625}{\omega^2}[1-\cos(2.72\omega)] \\ =\lim_{ \omega\rightarrow0^+}0.625(\dfrac{1-\cos(2.72\omega)}{\omega^2}) \ \ \ [0/0\ form] \\=\lim_{ \omega\rightarrow0^+}0.625(\dfrac{2.72\sin(2.72\omega)}{2\omega}) \ \ [L\ 'Hopital \ Rule]$

$\\=\lim_{ \omega\rightarrow0^+}\dfrac{0.625}2\times2.72^2(\dfrac{\sin(2.72\omega)}{2.72\omega}) \\=\dfrac{0.625}2\times2.72^2(1) \\=2.312$

Thus, we get same value, i.e., 2.312 from both methods.