Question

Prove that,the function f(x) =2x-|x| is continuous at the point x=0​

Answer 1

Answer:

The function f(x)=2x−∣x∣ can be written as

f(x)={2x+x2x−x::x≤0x>0.

Now,

x→0−limf(x)=x→0−lim3x=0.

And,

x→0+limf(x)=x→0+limx=0.

So we've,

x→0−limf(x)=x→0+limf(x)=f(0).

So the function f(x) is continuous at x=0.

Answer 2

Step-by-step explanation:

The function f(x)=2x−∣x∣ can be written as

f(x)={

2x+x

2x−x

:

:

x≤0

x>0

.

Now,

x→0−

lim

f(x)=

x→0−

lim

3x=0.

And,

x→0+

lim

f(x)=

x→0+

lim

x=0.

So we've,

x→0−

lim

f(x)=

x→0+

lim

f(x)=f(0).

So the function f(x) is continuous at x=0.