Question

Graph of the functions, cos (1/x) and sin (1/x) are weird because of the number of oscillations of the curve near the interval [-1 , 1 ].

Can you justify the reason for this densely oscillated curve?​

Answer 1

Step-by-step explanation:

The given functions works well in any number $\in[-1,1]$, except at $x=0$, due to the following reason:

$\bullet \: \sf{ \blue{As, \: \:x \to0 , the \: \: graphs \: \: of \: \: the \: \: given \: \: functions \: \: approaches \: \: its \: \: limiting \: \: value.}}$

Consider,

$\sf{ \lim_{x \to0} \bigg \{ sin \bigg( \dfrac{1}{x} \bigg)\bigg \} }$

$\sf{ = sin ( \infty ) }$

This implies, the graph could take any value $\in[-1,1]$. Even same value can also repeat many times because we can't say what value it will take.

Since $\sf{sin(\infty)}$ is not defined, but it can't exceed the range of sine function.

$\boxed{ \sf{\color{cyan} Hence,\:\:it\,\,oscillates\,\,in\,\,between\,\,[-1,1] } }$