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Show that lim(xrarr0)sin""(1)/(x) does n...

Show that `lim_(xrarr0)sin""(1)/(x)` does not exist.

Text Solution

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Let `f(x)sin""(1)/(x).`
`lim_(xto0)f(x)=lim_(hto0)f(0+h)=lim_(hto0)f(h)=lim_(hto0)sin""(1)/(h).`
Clearly, when h is given different values, then sin `1/h` oscillates between `-1 and 1 and ` thus it does not approach to a definite value. So, `lim_(hto0)sin""(1)/(h)` does not exist. Hence, `lim_(xto0)sin""(1)/(x)` does not exist.
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