Question

Does the limit exist at x=0 for the function f(x)=x+|x|/x?............. Please guys help me........ By solving.......

Answer 1

nope. The limit doesn't exist in this case because if we check the limit for 0+ ( For a value just greater than 0)
we will get:

x (+ve) + |x|(This value is always +ve as |x| is always +ve) /x(+ve)

you'll get : x + x/x = x+1

now,
If we check the limit for 0- ( For a value just less than 0)
we will get:

x (-ve) + |x|(This value is always +ve as |x| is always +ve) /x(-ve)

you'll get : x - x/x = x-1

here, as left hand limit isn't equal to the right hand limit ( x+1 is not equal to x-1)

the Limit doesn't exist.
hope this helps.

Answer 2

hey listen! limit exists when values approaching to limit are equal. means LHL & RHL must be equal for existence of limit. here your question. is( x+|x| )/x for x tending to 0+ . the value if limit will be (X + X )/ X i.e. 2 as |X| = X for X >0 and for x tending to 0- the limit will be (X–X)/X i.e 0 as |X|= –X for X<0 . since LHL & RHL are not equal . hence limit will not exist.