Question

Hey does anyone know if the answer is nonexistent?

Answer 1

Answer:

Non-existent is the answer.

Answer 2

Answer: Choice D is correct. The limit does not exist.

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Explanation:

Let's see what happens when we approach 0 from the left side.

So we could start at x = -1 and get closer to 0, but never actually get to that exact value.

If x = -1, then

$\frac{x}{|x|}=\frac{-1}{|-1|}=\frac{-1}{1} = -1$

Now let's try x = -0.5

$\frac{x}{|x|}=\frac{-0.5}{|-0.5|}=\frac{-0.5}{0.5} = -1$

You should find that any negative x value will lead to an output of -1.

You can make a table of values to help see this.

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In short, as x approaches 0 from the left, the y value will approach -1.

We would then write

$\displaystyle \lim_{x\to 0^{-}}f(x) = -1$

This is the left hand limit which I'll abbreviate as LHL.

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For similar reasoning, the RHL (right hand limit) is

$\displaystyle \lim_{x\to 0^{+}}f(x) = 1$

Because the LHL and RHL aren't the same value, this means that the limit at x = 0 does not exist. The two sides do not approach the same meeting point. It's like two cars, on two separate roads, that don't meet up. The roads need to connect.

Check out the graph below.

Side note: we avoid division by zero errors when we specifically state that x is nonzero for the first piece of that piecewise function.