Thread: Limits using absolute values

find limit if limit exists:

limit[x:-[infinity]](|x-2|+|3x|)/(x)

I assume that this is the question: $\displaystyle \lim _{x \to - \infty } \frac{{\left| {x - 2} \right| + \left| {3x} \right|}}{x}$.
Note that $\displaystyle x < 0 \Rightarrow \quad x = - \left| x \right|$. Which gives: $\displaystyle \frac{{\left| {x - 2} \right| + \left| {3x} \right|}}
{x} = - \left[ {\frac{{\left| {x - 2} \right| + \left| {3x} \right|}}
{{\left| x \right|}}} \right] = - \left[ {\left| {1 - \frac{2}
{x}} \right| + \left| 3 \right|} \right]$.
Now you finish.