1. Limits

I am having a problem with the absolute value problem when using limits. I don't know if I am right but I think that both of the questions lead to an answer of zero; however, I am not sure.

lim absolute value x
x->0

lim 1/absolute value x
x->0

2. You could compute $1/|x|$ for small values of x, e.g., 1/2, 1/100, 1/1000. This would show if the hypothesis that $\lim_{x\to0}1/|x|=0$ is plausible or not.

3. Originally Posted by happy8588
I am having a problem with the absolute value problem when using limits. I don't know if I am right but I think that both of the questions lead to an answer of zero; however, I am not sure.

lim absolute value x
x->0

lim 1/absolute value x
x->0
You should remember that

$|x| = \begin{cases}\phantom{-}x\textrm{ if }x \geq 0\\-x\textrm{ if }x<0\end{cases}$

1. To find $\lim_{x \to 0}|x|$, work out the left and right hand limits and see if they are equal. If they are, that is the limit. If they are not, the limit does not exist.

Approaching from the left means $x < 0$, so

$\lim_{x \to 0^-}|x| = \lim_{x \to 0^-}(-x)$

$= 0$.

Approaching from the right means $x > 0$, so

$\lim_{x \to 0^+}|x| = \lim_{x \to 0^+}x$

$= 0$.

Since the left and right hand limits are the same, the limit exists and is $0$.

2. Using the same logic, to find $\lim_{x \to 0}\frac{1}{|x|}$, check the left and right hand limits.

Approaching from the left means $x < 0$, so

$\lim_{x \to 0^-}\frac{1}{|x|} = \lim_{x \to 0^-}\frac{1}{-x}$

$= \infty$.

So this means that approaching from the left will make the function increase without bound.

Approaching from the right means $x > 0$, so

$\lim_{x \to 0^+}\frac{1}{|x|} = \lim_{x \to 0^+}\frac{1}{x}$

$= \infty$.

So this means that approaching from the right will make the function increase without bound.

Since both sides increase without bound, the limit is $\infty$.