Consider the following statements:
1. `lim_(xto0) (x^(2))/(x)` exists.
2. `((x^(2))/(x))` is not continuous at x=0
3. `lim_(xto0)(|x|)/(x)` does not exist.
Which of the statement given above are correct ?

Options

A. 1,2 and 3
B. 1 and 2 only
C. 2 and 3 only
D. 1 and 3 only

Answer

D

Explanation

`underset(xto0)lim(x^(2))/(x)=underset(xto0)lim(x)=0`
2. `(x^(2))/(x)=x`
since a polynomial is continuous everywhere, so it is continuous at x=0` `LHL=underset(hto0)lim(|0-h|)/((0-h))=underset(hto0)lim(h)/(-h)=-1`
`RHL=underset(hto0)lim(|0+h|)/((0+h))=underset(hto0)lim(h)/(h)=1`
`:'LHL""neRHL`
So, it does not exist.
Thus, the statements 1 and 3 are correct.