Justifying |lim f(x)| ≤ lim |f(x)| for continuous f

[mike1988]

I am trying to solve a question and I need to justify a line in which |lim(x-->0)(f(x))|≤lim(x-->0)|f(x)| where f is a continuous function.

Any help?

 

[micromass]

The absolute value is a continuous function. That is:

$$|\ |:\mathbb{R}\rightarrow \mathbb{R}:x\rightarrow |x|$$

is continuous. Does that help?? What do you know about continuity and limits?

 

[Ray Vickson]

I am trying to solve a question and I need to justify a line in which |lim(x-->0)(f(x))|≤lim(x-->0)|f(x)| where f is a continuous function.

Any help?

Show your work. Where are you stuck?

RGV

 

[mike1988]

Show your work. Where are you stuck?

RGV

actually I figured this out. Since || is a continuous, |lim(x-->0)(f(x))|= lim(x-->0)|f(x)| which is obvious from one of the theorems in my book.

Thanks though!