How do you find the limit of # (abs(x+2)-3)/(x-7 ) # as x approaches 7?

Answer 1

#lim_(x to 7^+) (abs(x+2)-3)/(x-7 ) = oo#

#lim_(x to 7^-) (abs(x+2)-3)/(x-7 ) = -oo#

#lim_(x to 7) (abs(x+2)-3)/(x-7 )#
#= (lim_(x to 7) (abs(x+2)-3))/(lim_(x to 7)x-7 )#

and because the numerator is continuous

#= (6)/(lim_(x to 7)x-7 )#

Note that this gives rise to a singularity and a 2 sided limit.

To test this in your head, mentally plug in, say, #x = 6.9# and #x = 7.1# So the right-sided limit is positive and the left-sided limit is negative

So we say that:

#lim_(x to 7^+) (abs(x+2)-3)/(x-7 ) = oo#
#lim_(x to 7^-) (abs(x+2)-3)/(x-7 ) = -oo#
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Answer 2

The limit does not exist.

Simple substitution of #x = 7# gives a numerator of 6 and a denominator of 0. There would be division of zero.
However, if #x# approaches 7 from the left, then the expression tends to negative infinity. We write
#lim_{x -> 7^-} frac{abs(x+2)-3}{x-7} = -oo#
Similarly, if #x# approaches 7 from the right, then the expression tends to infinity.
#lim_{x -> 7^+} frac{abs(x+2)-3}{x-7} = oo#
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Answer 3

To find the limit of (abs(x+2)-3)/(x-7) as x approaches 7, we can substitute 7 into the expression. However, this results in an undefined expression since the denominator becomes 0. Therefore, we need to simplify the expression before substituting. By factoring the numerator, we get abs(x+2)-3 = abs(x+2)-abs(3), which simplifies to abs(x+2)-abs(3) = abs(x+2)-3. Now, substituting 7 into the simplified expression, we get abs(7+2)-3 = abs(9)-3 = 9-3 = 6. Thus, the limit of (abs(x+2)-3)/(x-7) as x approaches 7 is 6.

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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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