Question

26. Find lim f(r), where /(x)-1x10,x→ 0x=0

Answer 1

Let the given function defined over range as

f( x )={ x | x | ,x≠0 0,x=0

We have to find the value of the given function at x→0 .

There are two different expressions for a given function defined at

x equal to 0 and other is for all values of x not equal to 0.

We need to take a common point at x=0

And find the left hand and right hand limit of the function.

From the definition of limits, we know that:

lim x→a f( x )=f( a )

So, on solving the expression for the value at x=0 ,

We know that modulus function | x |=−x , when x<0 and | x |=+x , when x>0 .

lim x→ 0 − f( x )= lim x→ 0 − x | x | = lim x→0 x | x | = lim x→0 −x x (When x is negative)

On applying limits:

lim x→0 −x x =(−1) (1)

Again,

lim x→ 0 + f( x )= lim x→ 0 + x | x | = lim x→0 x | x | = lim x→0 x x (When x is positive)

On applying limits:

lim x→0 x x =1 (2)

From equations (1) and (2), we found that

lim x→ a − f( x )≠ lim x→ a + f( x )

Thus, the value of the given function at x→0 does not exist.