help Finding the Value of limit

Question

I wanna the values of the limit :

$$\lim_{x\to 0}\frac{x}{|x|}$$

The answer is one of those ($\pm 1,+1,-1,$ undefined).

Answer 1

Let $x_n= \frac{1}{n}$ and we see that $ \lim_{n -> \infty} \frac{x_n}{|x_n|} =1$. Now consider $x_n= -\frac{1}{n}$ $ \lim_{n -> \infty} \frac{x_n}{|x_n|} = -1$. A sequence can only have 1 limit so the sequence does not converge.

Answer 2

The limit should be calculated from left and right of $0$ $$\lim_{x\rightarrow 0^+}\frac{x}{|x|}=\lim_{x\rightarrow 0^+}\frac{x}{x}=1$$ $$\lim_{x\rightarrow 0^-}\frac{x}{|x|}=\lim_{x\rightarrow 0^-}\frac{x}{-x}=-1$$

so the limit is $\pm 1$

Hence the limit not exit

Answer 3

You have to consider two limits:

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

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

Thus the limit is undefined.

Answer 4

The answer depends on whether you approach the value $0$ from the left or the right. Let's say that $|x| = \frac{1}{n}$ for some positive $n$.

Note that when we approach $0$ from the right $x=\frac{1}{n}$ is positive: $$ \lim_{x \downarrow 0} \frac{x}{|x|} = \lim_{n \to \infty} \frac{\frac{1}{n}}{\frac{1}{n}}=\lim_{n\to\infty}1 = 1.$$ On the other hand, if we approach $0$ from the left, $x = -\frac{1}{n}$ is negative, while $|x| = \frac{1}{n}$ is still positive: $$ \lim_{x \uparrow 0} \frac{x}{|x|} = \lim_{n \to \infty} \frac{-\frac{1}{n}}{\phantom{-}\frac{1}{n}}=\lim_{n\to\infty}-1 = -1. $$ So, in this case, the answer depends on whether you approach the limit from the left or the right. Therefore the limit in itself is undefined, as long as it is not specifically stated from which direction you approach the limit.