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

Question

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

Aaliyah Anderson · Accepted Answer

#31# values.

Given a domain of #[0.01, 1]#, #1/x# has a range of #[1,100]#. Thus, the question is equivalent to asking how many times #sin(x)# crosses the #x#-axis on the interval #[1, 100]#. As the graph of a function crosses the #x#-axis at the points where the function evaluates to #0#, and #sin(x) = 0 <=> x = npi, n in ZZ#, all that remains is to count the multiples of #pi# in the interval #[1,100]# We can verify that #0 < 1 < pi# and #31pi < 100 < 32pi#, meaning the only integers in which #npi in [1, 100]# holds are #n=1, 2, ..., 31#. As there are #31# such values, #sin(x)=0# has #31# solutions on #[1, 100]#. As this is equivalent to our original problem, we have that #sin(1/x)# crosses the #x#-axis #31# times for #x in [0.01, 1]#

Evelyn Arboleda · Answer

undefined 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.