# How do you find the limit of #|x+2| / (x+2)# as x approaches #-2#?

The limit does not exist.

We should determine if the limit from the left approaches the limit from the right.

Limit from the left:

Hence the limit from the left is

Limit from the right:

The function becomes

So the limit from the right is

Relating the limits:

graph{abs(x+2)/(x+2) [-13.77, 8.73, -5.62, 5.63]}

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To find the limit of |x+2| / (x+2) as x approaches -2, we can evaluate the limit from both the left and right sides of -2.

When x approaches -2 from the left side (x < -2), |x+2| simplifies to -(x+2) since x+2 is negative. Therefore, the expression becomes -(x+2) / (x+2), which simplifies to -1.

When x approaches -2 from the right side (x > -2), |x+2| simplifies to x+2 since x+2 is positive. Therefore, the expression becomes (x+2) / (x+2), which simplifies to 1.

Since the limit from the left side is -1 and the limit from the right side is 1, but they are not equal, the limit of |x+2| / (x+2) as x approaches -2 does not exist.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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