Consider the following statements:
1. `lim_(xto0)sin""(1)/(x)` does not exist.
2. `lim_(xto0)sin""(1)/(x)` exists.
Which of the above statements correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

B

Explanation

Since `sin""(1)/(x)` is an oscillatory function
`:.underset(xto0)limsin""(1)/(x)` has a finite value between- 1 and 1.
Now, At x=0
`L.H.L=underset(hto0)limf(0-h)=underset(hto0)lim(-h)sin(-(1)/(h))`
`=0xxa` finite value between -1 and 1=0
Similarly `R.H.L=underset(hto0)limf(0+h)=0`
Aslo, f(0)=0
`:.underset(xto0)limxsin""(1)/(x)` exists.
Hence, Both statement are correct.