A particle moving back and forth along a straight line has position function given by x(t)=sin(π(t−1)) with t in sec.

a) Estimate its instantaneous velocity at t=1 sec using a table of values (up to two decimal places).

What does your "table of values" say when t=1?.

(b) Using part (a), what can you say about the limit limx→0sin(πx)/x = Pi

There's no table of values given

A particle moving back and forth along a straight line has position function given by x(t)=sin(π(t−1)) with t in sec.

(a) Estimate its instantaneous velocity at t=1 sec using a table of values (up to two decimal places).

You have not given us a table so I will use calculus.

\(x(t)=sin(\pi(t-1))\\ \dot{x}(t)=\pi cos(\pi(t-1))\\ \dot{x}(1)=\pi cos(\pi(1-1))\\ \dot{x}(1)=\pi cos(0)\\ \dot{x}(1)=\pi *1\\ \dot{x}(1)=\pi\\ \dot{x}(1)\approx3.14\)

So instantaneous velocity when t=1 is  pi 

(b) Using part (a), what can you say about the limit limx→0sin(πx)/x ?

I am really not sure what you are being directed to do here but this is the answer.

Using L'Hopital's rule

\(\displaystyle\lim_{x\rightarrow 0}\;\frac{sin(\pi x)}{x}\\ =\displaystyle\lim_{x\rightarrow 0}\;\frac{\frac{d}{dx}sin(\pi x)}{\frac{d}{dx}x}\\ =\displaystyle\lim_{x\rightarrow 0}\;\frac{\pi cos(\pi x))}{1}\\~\\ =\pi \)