Graph the function f(x)= x sin( 1/x) and the equations y=|x| and y=-|x| in the same viewing window on your calculator. Then use the squeeze therom to find the limit of f(x) as it approaches zero.

Question

I graphed the function and equations, but I am confused on how to find the limit using the squeeze therom.

Answer 1

If you look at the graphs at

http://www.wolframalpha.com/input/?i=plot+x*sin%281%2Fx%29%2C|x|%2C-|x|

it's easy to see the squeeze.

for x>=0, x = |x| and |sin(1/x)| <= 1
so, |x sin(1/x)| <= |x|

x sin(1/x) is squeezed in between the |x| lines. So, while f(0) does not exist, it is between |0| and -|0|, or just zero.