# How many values of x between 0.01 and 1 does the graph #sin(1/x)# cross the x-axis?

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The graph of ( \sin(1/x) ) crosses the x-axis whenever ( \sin(1/x) = 0 ). Since the sine function equals zero at every integer multiple of ( \pi ), we need to find values of ( x ) such that ( 1/x = n\pi ), where ( n ) is an integer.

[ 1/x = n\pi ]

[ x = \frac{1}{n\pi} ]

For ( x ) to be between 0.01 and 1, we need ( n ) to satisfy:

[ \frac{1}{n\pi} > 0.01 ]

[ n\pi < 100 ]

[ n < \frac{100}{\pi} ]

Since ( n ) is an integer, the largest integer less than ( \frac{100}{\pi} ) is 31.

Thus, the graph crosses the x-axis for ( n = -31, -30, ..., -1, 1, ..., 30 ). So, there are ( 31 + 30 + 1 = 62 ) values of ( x ) between 0.01 and 1 where the graph crosses the x-axis.

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