How do you evaluate the limit #(abs(x+2)-2)/absx# as x approaches #0#?

Answer 1

The limit:

#lim_(x->0) (abs(x+2) -2)/abs(x) #

does not exist since the right and left limit are different.

Around #x=0# we have that: #x+2 > 0#, so #abs(x+2) = x+2#, and then:
#lim_(x->0) (abs(x+2) -2)/abs(x) = lim_(x->0) (x+2 -2)/abs(x) = lim_(x->0) x/abs(x)#

Evaluate separately:

#lim_(x->0^+) x/abs(x) = lim_(x->0^+) x/x =1#
As when #x->0^+# we have #x > 0# so #abs(x) = x#, while:
#lim_(x->0^-) x/abs(x) = -1#

We can conclude that:

#lim_(x->0) (abs(x+2) -2)/abs(x) #

does not exist since the right and left limit are different.

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Answer 2

To evaluate the limit (abs(x+2)-2)/absx as x approaches 0, we can consider the cases when x is positive and when x is negative.

When x is positive, the absolute value of x+2 is equal to x+2, and the absolute value of x is equal to x. Therefore, the expression becomes (x+2-2)/x, which simplifies to x/x, resulting in 1.

When x is negative, the absolute value of x+2 is equal to -(x+2), and the absolute value of x is equal to -x. Thus, the expression becomes (-(x+2)-2)/(-x), which simplifies to (-x-2-2)/(-x), resulting in (-x-4)/(-x), which further simplifies to (x+4)/x.

Since the limit is evaluated as x approaches 0, we can consider the limit from both the positive and negative sides. In both cases, the limit evaluates to 1.

Therefore, the limit of (abs(x+2)-2)/absx as x approaches 0 is equal to 1.

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Answer from HIX Tutor

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.

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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