Question

explain using linear approximations, why sin(x)≈x for x near 0

Answer 1

perhaps you have seen that
\[\lim_{x->0}\frac{\sin(x)}{x}=1\]that says the same thing

Answer 2

or you can use "linear approximation" aka find the equation for the line tangent to
\[\sin(x)\] at (0,0)
the slope is the derivative evaluated at 0. the derivative of sine is cosine, and cosine of 0 = 1 so the line has slope 1

Answer 3

and since it goes through (0,0) the equation for the line i s
\[y=x\] so that is the linear approximation for
\[\sin(x)\] near x = 0