Question

lxl25. Evaluate lim f(), where f(x)-1关x→00,x=0

Answer 1

Let the given function defined over range as

f( x )={ | x | x ,x≠0 0,x=0

We need to find the value of the given function at limit 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.