Analytical Limit

Limits, differentiation, related rates, integration, trig integrals, etc.

Analytical Limit

Hello, I'm having a problem with this limit, and any help you could all provide would be much appreciated.

Prove that if

$\lim_{x\to{c}}f(x)=L$,

then

$\lim_{x\to{c}}|f(x)|=|L|$.

Hint: Use the reverse triangle inequality

$|f(x)|-|L|\leq{|f(x)-L|}$

Posts: 1
Joined: Thu Dec 31, 2009 3:21 am

Re: Analytical Limit

AdvoTV wrote:Hello, I'm having a problem with this limit, and any help you could all provide would be much appreciated.

Prove that if

$\lim_{x\to{c}}f(x)=L$,

then

$\lim_{x\to{c}}|f(x)|=|L|$.

Hint: Use the reverse triangle inequality

$|f(x)|-|L|\leq{|f(x)-L|}$

http://en.wikipedia.org/wiki/Triangle_inequality#Reverse_triangle_inequality

$\left||f(x)|-|L|\right|\leq|f(x)-L|$

Use the standard $\epsilon,\delta$ definition of a limit.

Martingale

Posts: 350
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA